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Today we’ll see a puzzle that had me going in circles for longer than I’d like to admit.

It looks simple: two envelopes, one contains twice as much money as the other, and you get to pick one. Before you open it, I offer you the chance to swap. Sounds trivial—until you realize there’s a perfectly reasonable mathematical argument that says you should always switch, and an equally reasonable one that says it doesn’t matter at all.

Both arguments can’t be right. So which one is?

The Two Envelope Problem

Imagine you are given two sealed envelopes, each containing some money.

All you know is the following: one contains twice as much money as the other.

You pick one of them randomly. Before opening it, you are offered the change to switch.

What do you do?

Why You Should Switch

I’m going to convince you that you should switch.

Imagine the envelope you chose, let’s call it A, has 100€ inside. How much money is there in envelope B?

• If A is the envelope with the smaller quantity, then envelope B contains 200€ (recall one envelope contains twice as much money as the other one).

• If A is the envelope with the larger quantity, then envelope B contains 50€.

Since you chose randomly, half of the time you chose the envelope with the smaller amount, and the other half the one with the higher amount. Therefore, your expected value of switching envelopes is:

Note that the argument works if the first envelope had 200€ or any other amount inside it: if we call X the amount inside the first envelope you picked, then switching gives you an average payoff of 1.25X (which is higher than X).

The Infinite Loop

Okay, so we concluded that you are better off switching. You go and pick the other envelope.

Now here’s the catch: we are back at the start.

You have now picked an envelope, and we can apply the same argument again: it has an amount X, and, by switching, you get 1.25X on average (by the exact same calculation as before).

So you should switch again!

You can keep going forever, with math telling you that the best option is to keep switching infinitely: like a dog chasing its tail.

Why You Should Not Switch

Now here is a different argument that says you should not switch (or, rather, that you should be indifferent).

We know that one envelope contains twice as much money as the other one. Let’s say the quantities are 100€ and 200€.1

• If you picked the envelope with 100€ and you switch, you gain 100€ in profit.

• If you picked the envelope with 200€ and you switch, you lose 100€.

Since you picked the envelope randomly, half of the time we are in the first case, and the other half in the second. Therefore, your expected gain if you switch is:

So…Which Argument is Right?

Given these two arguments, what would you do?

The answer: you should be indifferent. Switching gains you nothing

The second argument is correct. The first argument–the one saying you should switch–is wrong, but spotting the flaw is surprisingly subtle. It involves something called improper priors, and a full technical explanation would take us into the weeds. But here's an intuitive version.

Go back to the critical step:

Imagine the envelope you chose, let’s call it A, has 100€ inside. […]

Since you chose randomly, half of the time you chose the envelope with the smaller amount, and the other half the one with the higher amount.

This quietly assumes that (50€, 100€) and (100€, 200€) are equally likely pairs. Fair enough. But the argument works for any starting number—so (1M€, 2M€) should be equally likely to (2M€, 4M€). And (1 trillion, 2 trillion) equally likely to (2 trillion, 4 trillion). And so on, forever.

But think about it: if you open your envelope and find 1 trillion euros inside, do you really think it’s 50/50 whether the other envelope contains 500 billion or 2 trillion? Of course not. Intuitively, the more money you see, the more likely it is you got the bigger envelope.

That intuition is mathematically correct. It turns out to be impossible for the pairs (X/2, X) and (X, 2X) to be equally likely for every possible X—the probabilities wouldn’t add up to 1. The first argument hides in an assumption about how the money was distributed that can’t actually be true. Once you account for realistic priors, the apparent 25% gain vanishes.

The dog stops chasing its tail.

Why This Matters Beyond Envelopes

The two envelope problem isn’t just a cute puzzle: it’s a warning. It shows how an expected-value calculation, done with apparent rigor, can give you an answer that is obviously absurd (switch forever!). The math feels right, but it’s quietly built on a flawed assumption about what’s equally likely.

This happens all the time outside of toy problems. Investment strategies, insurance bets, and even some published scientific results have fallen for similar traps: reasoning that looks clean on paper but rests on a prior that can’t actually hold up.

The lesson? When a calculation tells you there’s a free lunch, don’t reach for the fork. First, check the menu.

In Case You Missed It

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Game theory reveals a disturbing truth: rational self-interest can lead us to collectively terrible outcomes.

In the Prisoner's Dilemma, two criminals acting "optimally" for themselves both end up worse off than if they'd cooperated. This isn't just an academic puzzle: it explains why nations struggle to collaborate on climate change, why arms races spiral out of control, and why trust is so fragile in international politics. The mathematics are clear: when you can't trust your opponent, betrayal becomes rational, even when mutual cooperation would serve everyone better.

But there's hope: game theory also shows us exactly when and how cooperation can win.

The Prisoner’s Dilemma

In one of our first posts, we used the Prisoner’s Dilemma to introduce Nash equilibria—a central concept in game theory. Here’s a recap:

The Prisoners Dilemma

Two members of a criminal gang are arrested by the police. They are separated and can’t communicate with each other, and they have to go to court the next day.

The initial sentence is that both of them will be 1 year in prison. However, the police offer them individually the following deal: if one of them testifies against the other, they will not go to jail and, instead, the other will be sentenced to 3 years.

But, if both of them testify, the sentence of each will be 2 years instead of one.

What should they do to reduce their sentence as much as possible?

A Nash Equilibrium occurrs when, for any of the players, their strategy is optimal given the opponent’s one. In this game:

To be in a Nash Equilibrium, both players need to testify.1

Look at the outcome from this: both A and B get sentenced to 2 years. Had they cooperated and stayed silent, the outcome would’ve been better for both (each spending only 1 year in prison).

This is quite unsettling. Let me rephrase what is going on:

By not cooperating and looking after your own interest, you end up in a suboptimal situation.

From Prison Cells to Climate Change

This puzzle with prisoners and nice numbers has profound implications in the real world. Let me tweak the prisoner’s dilemma a bit and you’ll see why:

1. We don’t really need the two criminals to be separated and unable to communicate: as long as they don’t trust each other, the situation is effectively the same.

2. More general than “two criminals trying to reduce their sentence”, the dilemma appears in situations where two parties are trying to maximise some outcome, and their actions influence each other’s outcome (with a payoff with the same structure as the prisoner’s dilemma).

3. “Staying silent” becomes “cooperating”, and “testifying” becomes “defecting”.

Does this ring a bell? Here’s an example:

Climate Change

Two states A & B are facing the potentially catastrophic results of climate change.

If both take action to tackle the problem—they cooperate—the outcome is collectively the best: they each spend some money in a green transition, while growing on-par and reducing emissions together.

• Net outcome: (-1, -1)

However, if A works to reduce emissions while B doesn’t, A is at a clear disadvantage: imagine cleaning a river while a factory is polluting it upstream. Not only are you cleaning with no effect, but the factory owner is getting richer while you collect his shit. Furthermore, state B is in a better position than when cooperating: state A is doing the job for them, and they don’t need to invest anything in a green transition.

• Net outcome: (-3, 0)

Lastly, if neither of them takes action, they save money in the short term but will eventually face the consequences of climate change.

• Net outcome: (-2, -2)

What should the states do to reduce harm as much as possible?

And here’s the sad truth:

Just as in the prisoners dilemma, if you cannot trust your opponent, the best strategy is not to cooperate.

Can Trust Break the Trap?

Clearly things are not so simple in socioeconomic affairs.

Some countries trust each other and cooperate, while others don’t. These “cooperation partners”–our allies–are built through history by trial and error. Crucially, allies are time-dependent: someone can cooperate for some time, until they don’t.

Game theory has studied these time-dependent cooperative networks, and how to adapt strategies accordingly. For example, suppose you face the Prisoner’s Dilemma multiple times (with the same opponent), the so-called iterated prisoners dilemma.

This changes things dramatically: you can see how your opponent acts (do they cooperate or not?) and adapt your subsequent actions accordingly. The game becomes less about a single decision and more about building a reputation.

How would you act in such a situation?

Why Always Defecting Fails in the Real World

Interestingly, the Nash equilibrium in the iterated Prisoner’s Dilemma is still obtained by always defecting (which can be proved by induction). In other words, the math still says: betray, betray, betray

However, just as in the one-shot Prisoner’s Dilemma, the Nash equilibrium is not the optimal outcome for either party. If you trust your opponent and believe in cooperation, the best is to cooperate.

How should you act in a world where your opponent might cooperate?

Tit-for-Tat

This strategy was developed by Anatol Borisovich Rapoport, a mathematical psychologist (apparently such a profession exists). It was the best-performing deterministic strategy for the iterated Prisoner’s Dilemma in early computer tournaments.

It can be described with two simple rules:

1. On the first round, always cooperate.

2. On subsequent rounds, mimic your opponent’s last action: cooperate if they cooperated, don’t if they didn’t.

In other words:

Aim to cooperate, but retaliate if needed.

You start by trusting someone, and keep trusting them as long as they cooperate. However, you are no fool: if they stop being your ally and defect, you retaliate.

Conclusion

Two things can be concluded from this discussion.

First, it’s encouraging that mathematics tells us that, if cooperation and trust can be established, this is optimal. Always try to cooperate.

Second, the harsh truth is that, against someone who (consistently) defects, one must retaliate. Keep being the nice guy and you’ll just be exploited like a fool. As Luis & Pieter Garicano wrote on Sixteen thoughts on Greenland:

Even if ‘cooperating’ is less painful in the short run, against an opponent inclined to defect, the optimal response is retaliation. It is better to do this sooner rather than later […].

The world isn't a one-shot game.

We interact repeatedly: in diplomacy, business, and daily life. Tit-for-Tat shows us that the sweet spot lies between naivety and cynicism: start with trust, but don't be exploited.

In a world of repeated interactions, kindness with accountability isn't just morally appealing—it's mathematically correct.

In Case You Missed It

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What if I told you there's a new lottery you could play this Christmas?

Before you roll your eyes at yet another lottery scam, hear me out: this one is different. This lottery was invented by one of history's greatest mathematical minds, and for 300 years it has confused economists, philosophers, and mathematicians.

I'm going to show you this lottery, let you play it, and then ask you one simple question: how much would you pay to play?

Christmas Lotteries

Every Christmas, numerous countries around the world play their version of the Christmas lottery.

The prizes are mouth-watering. The Spanish Christmas lottery gives away a total of €2,702M; and if you happen to win El Gordo—the biggest prize—you get €400,000 for each décimo (€20) that you play. The Lotteria Italia has a much smaller total pool (around €21M), yet the first prize rises to an astonishing €5M.

Any sane person would like to win El Gordo or the Lotteria Italia. And, indeed, millions of people pay the €20 that it costs to buy a décimo, or the €5 that it costs to play the Lotteria Italia.

Now here’s the catch. Despite its large prizes, you’d probably not play the Spanish Christmas lottery if the prizes were the same but you had to pay €100 for each décimo. It wouldn’t be worth the price.

But what if I showed you a lottery that mathematically should be worth any price—yet no one would pay more than €20 to play?

The St. Petersburg Lottery

This Christmas, imagine the city of St. Petersburg hosting the strangest lottery you've ever heard of.

Instead of drawing numbers, in St. Petersburg they toss a coin. Here’s how it works:

• If tails comes up on the 1st toss, your prize is €2.

• If tails comes up on the 2nd toss, your prize is 4€.

• If tails comes up on the 3rd toss, your prize is 8€.

• And so on—doubling each time.

The game continues until the first heads appears, at which point you collect your prize and go home.

Go try it! You can play the St. Petersburg lottery here.

Now here’s my question for you:

The Paradox

If you try playing the St. Petersburg lottery a few times, you won’t be too impressed with the prizes.

Half of the time you get paid only €2. Only around 6% of the time you’ll get a prize of 32€ or more. And the odds decrease exponentially:

• You get more than €100 only 1.5% of the time

• You get more than €1,000 only 0.2% of the time

• To equal El Gordo and get €400,000, you’ll need to play this lottery over 260,000 times!

The St. Petersburg lottery was invented by the Swiss mathematician Nicolaus Bernoulli around 1713. According to Bernoulli himself:1

Any fairly reasonable man would sell his chance, with great pleasure, for ten euros.

I’m guessing most of you will be willing to pay somewhat in the range €5-20 to play this game, just as Bernoulli suggested.

But here is the shocking truth:

On average, the St. Petersburg lottery will pay you infinite money.2

The expected payoff of the lottery is infinity, thus you should be willing to pay any amount of money to play it!

The fact that we are willing to pay only ~€10 in a game that has an expected value of infinity is known as the St. Petersburg paradox. Let’s explore why this lottery feels worth much less than its mathematical value suggests.

Solution #1: It’s About Utility, not Euros

One solution—interestingly, due to Nicolaus’ cousin Daniel Bernoulli—involves utility theory.

We explained in an old post that utility theory is the mathematical theory that explains the value we assign to money. The same €1,000 doesn’t have the same value to someone who has €100 in savings as someone who has €10,000.

The more money we have, the less we value each additional unit.

It looks something like this:

Instead of asking “What is the expected value of the St. Petersburg lottery?”, Daniel Bernoulli’s utility theory says we should ask “What is the expected utility gain from the St. Petersburg lottery?”.

Using a (natural) logarithmic utility, here’s what we get:

• A person with €1M in savings should pay up to €20.88 to play this game.

• A person with €1,000 in savings should pay up to €10.95.

• A person with €2 should borrow €1.35 and pay up to €3.35.

This makes sense: the likely outcomes of the St. Petersburg lottery (winning a small amount of money) are a significant wealth increase for someone with €2, making it profitable for them to risk a high percentage of their savings, but not for someone with €1M.

Solution #2: Our Brain Rounds Down to Zero

Another explanation, due to Nicolaus himself, is that we neglect sufficiently small probabilities.

There is a chance that you’ll get a prize of over €1M—but its likelihood is 1 in 524,288 (you need to play 524,288 times on average to see such prize). Nicolaus argued that we humans perceive such tiny probabilities as literally zero.

In other words, Nicolaus’ point is that, although the game states that we play until we get a heads (and at that point we take the prize), we humans perceive the game effectively as:

Toss a coin until the first heads appears, but at most you’ll toss it X times.

After a point, we just perceive the outcome to be impossible.

If our brain unconsciously caps the game at, say, 20 tosses, the expected value drops from infinity to just €20—which matches our intuition perfectly.

What the St. Petersburg Paradox Teaches Us

The St. Petersburg paradox reveals a profound truth: human decision-making isn’t only about mathematics—it’s about psychology, risk perception, and the subjective value of money. Whether it’s Daniel Bernoulli’s utility theory or Nicolaus’s probability neglect, both solutions tell us the same thing: we’re not irrational for refusing to pay €1,000 for this lottery. We’re human.

So this Christmas, when you see those lottery tickets, remember: the expected value isn’t everything. Sometimes, the smartest bet is the one that lets you sleep at night—even if the math says otherwise.

In Case You Missed It

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Charlie Munger once said “All I want to know is where I’m going to die so I’ll never go there”.

When imagining your death, you probably think of a dramatic, sudden event: a deadly car crash or some lethal rare disease. The reality is actually very different. Unless you are a professional base jumper or practitioner of a high-risk activity, you’ll die of a chronic disease of aging.

Today we will explore Outlive: The Science & Art of Longevity, by Dr. Peter Attia: a novel, prevention-based approach to health and lifespan that will change the way you approach your health.

The Four Horsemen & Medicine 3.0

Unless you regularly practice a risky activity, you’ll likely die from one of the four chronic diseases of aging:

1. Heart disease.

2. Cancer

3. Neurodegenerative disease

4. Type II diabetes (and related metabolic dysfunction).

Attia calls these the four horsemen, in analogy to the four horsemen of the apocalypse. To defeat them, he introduces the concept of medicine 3.0:

• Medicine 2.0 (traditional medicine) is the kind of medicine that we are used to. In medicine 2.0, you go to the doctor to receive medication or treatment when something is wrong.

• In Medicine 3.0, the focus shifts from intervention to prevention. We want to stop the diseases before they develop. And we want to live better, not just longer.

The critical insight?

You are no longer a passive observer. Your life, your hobbies, and your daily choices are the ultimate longevity drug.

We must shift our objective (healthspan and lifespan) into a clear strategy (prevention over cure), and then execute with tactics (Exercise, Nutrition, Sleep, and Emotional Health).

Let’s explore these four tactics.

Exercise: The Most Potent Longevity Drug

Out of all the things you can do to improve your health and lifespan, there is a king: exercise.

No other intervention is nearly as effective as exercise to prolong lifespan and preserve cognitive and physical function.

Regular exercisers live up to a decade longer than sedentary people. And not just longer, but also better.

To maximize these benefits, you must master four main categories of fitness:

1. Aerobic Endurance and Efficiency (Zone 2)

Zone 2 cardio is steady-state training that builds mitochondria’s health. It teaches your body to utilize fat as fuel more efficiently, which is key for metabolic health and cognition.

• What is it? Effort where you can just hold a conversation (light jog, uphill hike).

• How much? Aim for about 3 hours per week.

2. Maximum Aerobic Output (VO2 Max)

This is about short, high-intensity efforts that push your body’s maximum rate of oxygen consumption.

Your VO2 max is the single most powerful marker of longevity.

• What is it? VO2 max training involves intense intervals, like 4 minutes as fast as you can followed by 4 minutes of full rest.

• How much? You really don’t need a lot: 1 session per week is sufficient.

VO2 max naturally declines with age, but you have a lot of power to slow it down.

3. Strength

The longevity benefit of lifting is the retention of functional strength.

We lose strength 2-3 times faster than muscle mass. A surprising predictor of overall health is your grip strength.

• A good place to start: achieve the farmer’s walk—carrying roughly half your body weight in each hand for one minute.

4. Stability

The most critical factor for sustaining a long healthspan is avoiding injury.

Stability training ensures that the force you generate is transmitted through muscles and bones, not dispersed along fragile joints.

• The Goal: Build a body that can withstand the stress of an active life, often beginning with conscious breath work and core control.

Nutrition: Getting Your Metabolic House in Order

The first step in delaying death is to get your metabolic house in order. This is where diet and insulin play a crucial role.

The Nature of Metabolic Syndrome

Nearly half the US population is metabolically unhealthy or on track to be.

Metabolic syndrome is defined by meeting three or more of five risk factors, including high blood pressure, high triglycerides, and elevated fasting glucose.1

When you are metabolically healthy, extra calories (like that cookie) are stored in your subcutaneous fat (the fat under the skin). But once those fat cells are full, and you keep eating and don’t move, the fat spills over. It ends up in your muscles, in your liver (NAFLD—Non-Alcoholic Fatty Liver Disease), and in your visceral fat (the dangerous fat wrapped around your organs).

This process—driven by insulin resistance—is the common thread linking Type 2 diabetes, heart disease, and even Alzheimer’s.

Protein is Your Building Block

The most common mistake is undershooting protein.

Standard recommendations (around 0.8 g/kg/day) are a joke; muscle loss has been observed at this rate. You need significantly more, especially as you age.

Aim for 1.6 g/kg/day minimum, and optimally closer to 2.2 g/kg/day of body weight.

Basic Rules of Nutrition

When it comes nutrition, there is very little we know with certainty. However, there are som well established rules that we should try adhering to:

1. Calorie intake: eat just enough calories (not too many, not too little).

2. Eat enough protein and essential fats

3. Eat a diet rich in minerals and vitamins

4. Avoid pathogens and toxins (mercury, led…)

Nutrition is a really complicated topic, and you can hear very opposing views (keto is best! No, it’s fasting! You should eat curcumin!). However, Attia emphasizes to stop overthinking nutrition so much and get exercise instead.

Sleep: The Best Brain Medicine

We often wear chronic sleep deprivation as a badge of honor (”I only need five hours!”). This is dangerous bullshit. People who are sleep-deprived underestimate the effect because their baseline has reset to a suboptimal state.

Sleep deprivation is horrible for insulin resistance. The evidence is crystal clear.

• Quantity: Target 7.5−8.5 hours. Both too little and too much are problematic.

• Quality: Poor sleep puts your body in a state of alert, increasing blood pressure and heart rate. It literally increases your risk of a heart attack.

Sleep is when the brain gets its cleaning crew:

• Deep sleep clears short-term memory and selects what to store.

• REM sleep is crucial for problem-solving, creativity, and processing emotions.

If you want to prevent neurodegenerative diseases, good sleep in middle age (40s to 60s) is non-negotiable.

The Golden Rule: Fix your wake-up time and don’t deviate, not even on weekends. Budget for at least 8 hours in bed every night.

Some tips for better sleep

1. Rule out problems like sleep apnea.

2. Take care of your environment: keep your room completely dark and cool (about 18°C).

3. Limit alcohol.

4. Stop caffeine at least 6-8 hours before bed.

5. Mentally prepare for sleep (be relaxed: no email, social media or news before bed).

6. No food 3h before going to bed.

   

Emotional Health: Why Does Any of This Matter?

The final, and perhaps most philosophical, vector of health is emotional health.

Longevity is meaningless if your life or your relationships suck.

Attia points out that an unhappy life is simply waiting for death. Living or feeling alone is strongly linked to a higher risk of mortality.

Your physical health plan must be twinned with an emotional strategy that helps you deal with past trauma, build functional relationships, and cultivate mindfulness—the ability to detach yourself from your immediate emotions, creating a gap between stimulus and response.

People get old when they stop thinking about the future. If you want to know someone’s true age, don’t look at their skin—listen to them. If they talk about their dreams, aspirations, and what they are looking forward to, they are young. If they only talk about the past, they’re old.

The Good News: It’s In Your Hands

Genes play a huge role in determining your health and life span, and there is nothing that we can do to change them. But our behaviours can bridge the gap.

Can we, through our behaviors, rip the same benefits that centenarians get for free?

The answer is yes.

In Case You Missed It

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Why do we sometimes buy lottery tickets we know we’ll lose, and at other times pay for insurance we know will never “pay back”?

At first glance, these two behaviors seem opposite: one is chasing risk, the other is avoiding it. Yet both come from the same psychological quirk—how our minds perceive possibility.

This post explores how we assign value to money and uncertainty. Through a few simple examples, we’ll see why our attitude toward risk flips depending on whether we’re thinking about gains, losses, or unlikely events. And how the “possibility effect,” a key idea from Daniel Kahneman’s Thinking, Fast and Slow, quietly shapes decisions we make every day.

Money vs. Utility

We have talked about utility several times in this newsletter.

If you haven’t read those posts, here’s a quick reminder:

Utility is the value we assign to money.

You don’t value 100€ the same way if you have 200€ in your account as when you have 2,000€. Same money, different utility.

Everyone has their own utility function, but most of us share some general patterns. And these patterns have huge implications for how we deal with risk.

Securing Gains

For most people, utility grows slower as we have more money—economists call this a concave utility function for gains.

Here’s what that looks like in practice.

I offer you two deals:

1. Tomorrow I give you 50€.

2. Tomorrow we flip a coin.

   1. If it lands heads, you get 110€.

   2. If it lands tails, you get nothing.

Most people pick the sure 50€. That’s because we’re risk averse: we prefer a guaranteed profit over an uncertain one, even if the uncertain one has a higher expected value.

Risk aversion is the tendency to prefer certain outcomes over uncertain ones, even if the uncertain ones have a better expected value.

Fearing Losses

But when losses enter the picture, something strange happens.

Let’s try another question:

What would you choose?

1. Lose 900€ for sure.

2. A 95% chance to lose 1,000€ (and a 5% chance to lose nothing).

Now, most people pick the gamble. They become risk seeking.

Why? Because the chance of losing nothing feels huge compared to the extra 100€ you might lose. Losing 900€ already hurts so much that losing 1,000€ doesn’t feel much worse.

The pain of losing 900€ is stronger than 95% of the pain of losing 1,000€.

This flip—risk averse for gains, risk seeking for losses—is one of the key insights of prospect theory.

Where Things Get Tricky

Prospect theory summarizes this behavior with an S-shaped utility curve— concave for gains, convex for losses.

But when probabilities get tiny, things change again.

Dreaming Big

Imagine this:

I offer you two options:

1. Get 120€ now.

2. A 1-in-1,000 chance to win 100,000€.

Most people go for the gamble. The logic is the same as why we play the lottery: without a ticket, you can’t win; with a ticket, at least there’s a chance.

As Kahneman puts it:

“A lottery ticket is the ultimate example of the possibility effect. Without a ticket you cannot win, with a ticket you have a chance, and whether the chance is tiny or merely small matters little.”

This is the possibility effect: our minds overweight very small probabilities, making rare events feel more likely than they are.

Whereas the concavity of the utility function pushes us to be risk averse and prefer sure outcomes over uncertain ones, the possibility effect pulls in the other direction and makes us be risk seeking.

When facing unlikely but life-changing gains, the possibility effect makes us risk seeking—the opposite of our usual behaviour for gains.

Fearing the Worst

Now flip the situation.

Would you rather:

1. Lose 110€ now.

2. Take a 1% chance to lose 10,000€.

This one sounds familiar, doesn’t it?

Every time you buy insurance—for your phone, car or home—you are choosing the small, certain loss over the very unlikely large one.

If you do the math, the gamble (option 2) usually has a better expected value (and expected utility!). Yet most of us pick the sure loss. Why?

Again, the possibility effect. Our brains overweight that tiny 1% chance of catastrophe. Even if the math says it’s irrational, the feeling of safety is worth paying for.

In other words, we’re not buying financial protection—we’re buying peace of mind.

It doesn’t matter how small our chance of a big loss is—what matters is that there is a chance.

Conclusion

When we face uncertainty, two psychological forces pull us in opposite directions.

• The shape of our utility function makes us risk averse for gains and risk seeking for losses.

• The possibility effect makes us overestimate small probabilities, becoming risk seeking for unlikely gains and risk averse for unlikely losses.

Together, they explain why we both play the lottery and buy insurance.

You might play because you’ve fallen for the possibility effect—or simply because dreaming is fun. You might buy insurance because you overvalue that 1% risk—or because you genuinely want peace of mind.

Either way, the point isn’t to change how you act. It’s to understand why you act that way—and to make sure you’re the one choosing, not your biases.

In Case You Missed It

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Imagine you’re at a store and see a phone that costs 700€. You manage to get a 15% discount—how much are you saving?

If you reach for your calculator, you are not alone. Most of us do. But what if you could calculate it in your head—fast, confidently, and even enjoy it?

Mental math isn’t about showing off or memorizing tables. It’s about learning to see numbers differently—spotting shortcuts and patterns that make calculations simple. Let’s explore a few tricks that will make math feel lighter, quicker, and surprisingly fun.

The Fast Way

The easiest way, of course, is to use a calculator.

Type in 0.15 × 700 and you’ll find out you’re saving 105€.

And honestly, I’d recommend this in most cases—if a calculator can do it for you, don’t waste brainpower! You’ll get a faster and more reliable answer.

But sometimes you don’t have a calculator—or you just want to sharpen your mental math. That’s when a few tricks come in handy.

The Back-Up Plan

Without a calculator, you could multiply 0.15 × 700 the usual way—by hand or on paper. It’ll work, but there’s a smarter way.

Let me share a few techniques I personally find useful.

1. Forget the Zeros

Whenever you multiply numbers ending in zeros, ignore the zeros first and add them back later.

Example: 25 x 20
25 x 20 → ignore the zero in 20 → 25 x 2 = 50 → add one zero back → 500.

This trick also works for larger numbers:

• 120 × 300 → compute 12 × 3 = 36 → add four zeros → 36,000.

• 2,380 × 200,000 → compute 238 × 2 = 476 → add six zeros → 476,000,000.

This reduces clutter and helps you focus on the meaningful digits.

2. Divide & Conquer (Multiplication)

Split big numbers into smaller, easier ones.

Example: 12 x 25
12 × 25
We can write 12 = 6 × 2, so 12 × 25 = 2 × (6 × 25) = 2 × 150 = 300.

Another version: when multiplying by 5, remember 5 = 10 ÷ 2.

Example: 5 x 44
5 × 44 → divide 44 by 2 = 22 → then multiply by 10 → 220.

3. Divide & Conquer (Addition)

You can do something similar with addition.

Example: 147 + 234
- Round both numbers slightly: 147 = 150 - 3, and 234 = 230 + 4.
- Now add (150 + 230) +(- 3 + 4).
- Put everything together 147 + 234 = 381.

Breaking numbers into round figures makes the mental math smoother and reduces mistakes.

4. The Distributive Power — Share Everything

Here’s the secret weapon: the distributive property.

It says that multiplying a sum is the same as summing the products:

Since adding is easier than multiplying, this property is gold.

Example: 12 x 25
- Write 12 = 10 + 2.
- (10 + 2) x 25 = (10 x 25) + (2 x 25) = 250 + 50 = 300.

Simple, right? This property lets you handle big numbers by breaking them into friendlier chunks.

Be Creative!

Once you get comfortable, you can start mixing tricks.

Example: 0.15 x 700
- Write 0.15 = 0.1 + 0.05.
- 0.1 x 700 = 70
- 0.05 x 700 = (0.1 x 70) / 2 = 35.
- Add them: 70 + 35 = 105.

Example: 34 x 90
- Forget the 0 in 90 → think of 34 x 9, and add a zero later.
- Write 34 = 30 + 4.
- 30 x 9 = 270
- 4 x 9 = 36.
- 270 + 36 = 306.
- Add the zero →3,060.

Example: 9 x 78
- Write 9 = 10 - 1
- 10 x 78 = 780.
- 1 x 78 = 78.
- Subtract → 702.

Conclusion

Mental math isn’t about doing everything by hand — it’s about understanding numbers well enough to play with them.

In fact, this is how we mathematicians do mental math. We’re not faster at multiplying digits one by one—we’re better at seeing structure: splitting numbers, spotting patterns, and using properties like the distributive rule. The secret isn’t speed—it’s strategy.

The shortcuts above not only make calculations faster; they build intuition. After a while, you’ll start to “see” numbers differently—round them, split them, recombine them—and realize that what once felt like arithmetic is really just pattern recognition.

So next time you get a discount, or split a bill, try one of these tricks. You might be surprised at how fast your brain catches up.

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Why do small towns have the highest cancer rates? And why do polls get more accurate as they grow?

Last week, we met the Law of Large Numbers—the mathematical principle behind why polling more people gets us closer to the truth. This week, we’ll go a step further. We’ll not only ask how close we can get, but how sure we can be.

This is the story of the Central Limit Theorem—a law that hides behind election forecasts, scientific studies, and that explains how averages of random samples behave. It’s one of the most powerful and beautiful ideas in all of probability, and by the end of this post, you’ll understand it intuitively.

Measuring Cancer

Let’s start with a little thought experiment.

Imagine you want to study the incidence of lung cancer across different parts of your country. To do this, you consider 1,000 different cities in your country and, for each of them:

1. Count how many people have (or had) lung cancer.

2. Divide this count by the total population of the city.

The resulting number is called the rate.

You find something curious: among the 10 cities with the lowest rate of lung cancer, 8 of them have a population of less than 2,000 people.

What would you conclude from this study?

• A. Smaller cities are probably remote and have better air quality.

• B. Among the cities with the highest rate of lung cancer, most also have less than 2,000 people.

• C. Smaller cities have healthier lifestyles.

My guess is that most people will pick either A or C — and think that B must be wrong. If that’s you, congratulations: you’ve just fallen for the narrative fallacy.

Your brain sees a result and instantly invents a story. It looks for a plausible cause to match the effect.

But here’s the truth:

What you are seeing is the mere effect of randomness.

Election Day

Let’s return to a familiar example: election night. Pollsters survey a few thousand voters and make predictions about who will win.

With only 1,000 people surveyed (green), the reported support for a candidate might not match the true result. When the sample is larger —say 6,000 voters (yellow)—the prediction is usually much closer to reality.

This aligns with something you might already sense intuitively:

The more people we poll, the closer we get to the true value—and the more confident we can be.

The first part comes from the Law of Large Numbers:

Poll enough people, and you can get as close as you want to the truth.

The second part is what the Central Limit Theorem (CLT) explains:

It tells us how confidence increases as we poll more people—and by how much.

The Central Limit Theorem

Let’s state it formally, and then unpack it in plain language.

The Central Limit Theorem

Let X1, …, Xn be independent and identically distributed random variables with mean E and finite variance V.

Then their average converges in distribution to a normal random variable with mean E and variance V/n.

Let’s unpack what this is saying.

Breaking It Down

Imagine you want to know what is the average height of the Spanish population.

You call people at random, ask their height, and compute the average. This is precisely the set-up of the Central Limit Theorem.

The theorem assumes we have random quantities X1, …, Xn— independent and identically distributed—each with mean E and variance V.

In plain terms:

• X is the height of a random person.

• E is the true average height (which we don’t know).

• V measures how much individual heights vary.

The larger the variability in your observations, the larger the variability of your estimate.

The conclusion

The CLT says that the average you report follows a normal distribution centered at the true mean E, with variance V/n.

Note that the average you report is random: when you run the experiment again you will collect the heights of a different group of 1,000 people, so the average height you report will be different.

If you repeat the experiment many times and plot the averages, this is what you’ll see:

The CLT says that, as you increase the number of people you poll:

1. The blue histogram gets closer and closer to the red dashed line.

2. The shape becomes thiner and taller.

When the number of people we poll is small (n=30, top left), it’s still easy to get 174 or 178 cm even if the true mean is 176 cm.

However, once the number of people we poll is large (n=1000, bottom right), almost all averages are clustered tightly around 176 cm.

The Original Puzzle

Now we can return to the lung cancer example. Hopefully now you can see that the correct answer is B.

Think of a town with only one person: if they have cancer, the rate is 100%; if not, 0%. You’ll never see such extremes in a city like Madrid.

In small towns, random fluctuations have a huge effect — and that’s why extreme values are more likely to appear in small populations.

The Lesson

The Central Limit Theorem explains why randomness itself tends to look orderly at scale.

No matter what we measure—heights, votes, or diseases—if we average enough independent samples, the chaos smooths out into a perfect bell curve.

That’s the magic of the CLT: it shows that certainty can emerge from uncertainty, and that behind the noise, there’s always a pattern waiting to be discovered.

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Imagine this: it's election night, and pollsters declare a winner based on polling just 2,000 voters out of several million. Sounds impossible, right? Yet these predictions are close to the results with remarkable accuracy.

The secret isn't dark magic or crystal balls—it's one of statistics' most elegant principles at work. The Law of Large Numbers explains why small, carefully chosen samples can reveal profound truths about massive populations. It's the mathematical foundation that makes opinion polls, medical trials, and quality control possible.

Today, we'll explore this statistical wonder and discover why "going big" with numbers is our most reliable path to truth.

The Mystery: How Do Pollsters Do It?

Let’s start with a concrete example

Suppose you want to know what percentage of the population in Spain will vote for PSOE in the next election. That percentage exists as a fixed number somewhere out there—if we could magically ask every single Spanish voter, you could calculate it perfectly. The problem? You can’t feasibly ask 37 million people.

Instead, you do something that seems almost too good to be true: you ask a few thousand people, calculate their percentage, and declare that this represents the entire country.

This extrapolation from sample to population turns out to be not wild at all—it's precisely what the Law of Large Numbers guarantees will work.

The Law of Large Numbers: Your Statistical Superpower

Here's the formal definition, then we'll break it down:

The Law of Large Numbers

The average of a sample of independent, identically distributed random variables converges to the population average as the sample size increases.

Sounds intimidating? It's actually describing something really simple: poll enough people the right way, and you'll get remarkably close to the truth.

Let's unpack each part:

Random Variables: When Certainty Meets Chance

A random variable is just the name mathematicians use for "a quantity that varies randomly."

Here's the key insight: whether María specifically votes for PSOE isn't random—she knows her preference. But if I pick someone at random from Spain's population, their preference is random from my perspective. I genuinely don't know what I'll discover until I ask.

Independence: Why Your Friends Can Ruin Everything

Here's where things get tricky, and why many polls fail.

Imagine you're estimating the support for PSOE, so you poll someone and also their partner. Problem: couples often share political views. If María votes PSOE, Carlos probably does too.1

This creates "dependent" responses—they carry overlapping information rather than independent evidence. It's like asking the same person twice and pretending you got two opinions.

With strong dependence, the Law of Large Numbers can break down entirely. Positive correlations make your sample mean jump around more, requiring much larger samples to stabilize.

Identically Distributed: Comparing Apples to Apples

This is just a statistician’s way of saying that you should "make fair comparisons."

Want to know the average height of Spanish men? Sample Spanish men and measure their height—not Greek men, not Spanish women, and don’t measure their weight instead.

Most importantly, your sample must be representative: measure only people in Galicia, and you're measuring Galicia’s average height, not all of Spain's.

Convergence: Getting Arbitrarily Close to Truth

Remember when we learned that 0.9999… = 1? The sequence 0.9, 0.99, 0.999… gets as close to 1 as you want.

The same happens with polling averages. "Convergence" means:

Poll enough people, and you can get as close to the true answer as you want.

The LLN in Action: A Real Simulation

Let me show you the Law of Large Numbers in action with a real simulation.

Imagine we somehow know that exactly 32% of Spain supports PSOE (roughly their 2023 election result). Now let's watch what happens as we poll 1, 2, 3… up to 8,000 random people:

As you can see in the figure, at the beginning our estimate is very poor.

After polling just one person (who happened not to support PSOE), our estimate is 0%—obviously terrible.

As we poll more (but still few) people, our estimate of the support for PSOE goes up and down erratically, reaching almost 40% in early stages and dropping below 30% with around 1,000 people polled.

The crucial thing is that, as the number of polled people increases, you can see how our estimate of the support for PSOE stabilises around the true value. This means two things:

• It doesn’t vary much.

• It stays close to the true value.

After about 2,000 people polled, the sample average stays within a 1% range of the real value of 32%.

If you run another poll with 8,000 new people, you will get a different graph, yet the phenomenon will be the same: as we poll more and more people, the sample average stabilises and approaches the true mean.

Why This Changes Everything

The Law of Large Numbers isn't just mathematical theory—it's the invisible foundation of our data-driven world.

It explains why pharmaceutical companies can test drugs on thousands of people and confidently predict effects on millions. Why manufacturers can inspect a few hundred products and guarantee quality across entire production runs. Why Netflix can recommend movies based on user patterns, and why your bank can assess credit risk.

The next time you ponder about the power of a poll, remember that mathematics guarantees that asking enough people reveals the collective truth.

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Every day, we make countless unconscious decisions—each hiding a potential mathematical puzzle.

When it starts raining as you walk home and you start running, does that really keep you drier? When you accept an offer simply because "it seems high enough", is that truly the optimal thing? We often follow instinct—doing what "feels right".

Let me show you how mathematics can even help with something as mundane as tying your shoe.

The Travelator Problem

Here’s an interesting puzzle posed by the mathematician Terence Tao in his blog:

You are at the airport and need to get from Terminal A to Terminal B.

Suppose you walk at a constant speed. Along the way are moving walkways (“travelators”) that add their own speed to yours.

As you start walking you realise that you need to tie your shoe. You have to decide whether to stop on or off a travelator.

If your goal is to get to Terminal B as quickly as possible, where would you stop?

My First Thoughts

When I first encountered this problem, my gut told me I should tie my shoe while on the travelator.

I couldn’t justify it rigorously, but I thought: if I stop on the travelator, "at least I’m moving a bit while tying my shoe."

The travelator problem was presented in The Guardian in 2016. Most people shared my gut feeling:

But after thinking it through, I began to suspect it didn’t matter where I stopped.

My reasoning was along the following lines. Let’s say I walk at a constant speed of 4 km/h, and the travelator moves at a constant speed of 3 km/h.

• If I stop walking on the travelator, I’ll move at 3 km/h instead of 7 km/h. So I’ll move 4 km/h slower while I tie my shoe.

• If I stop walking off the travelator, I’ll move at 0 km/h instead of 4 km/h. Again, 4 km/h slower while I tie my shoe.

So, regardless of where I decide to tie my shoe, I’ll just be moving 4 km/h slower during that time.

Then I wondered whether the answer depends on the exact speeds. Maybe the math would get messy…

Solving the Puzzle

The Hard Way—Doing The Math

One way to solve the puzzle is by simply doing the math.

Let u be your walking speed and v the travelator’s speed. You can set up and solve a simple system of equations.

Here is what you will find:

Regardless of what your walking speed and the travelator’s speed are, it’s always best to stop on the travelator.

You’ve done the math, so this is the answer. If you have any curiosity at all, you’ll probably be wondering:

But why?

The Simple Way—A Beautiful Solution

Here is an elegant argument that shows why it’s best to stop on the travelator.

Imagine two people, Alice and Bob, both walking from Terminal A to Terminal B. They both need to tie their shoe along the way. As they approach the first travelator:

• Alice stops just before the travelator to tie her shoe.

• Bob stops immediately after stepping on the travelator to tie his shoe.

Now see what happens.

While they tie their shoes, Bob keeps moving and Alice doesn’t. When they start walking again, Bob’s head start is permanent—he reaches Terminal B first.

A Challenge for You

Terence Tao also proposed the following problem, which I will leave for you to ponder on:

If you can run for some time instead of walking, where would you run?

Takeaways & Everyday Applications

In this puzzle, our instinct—stop on the travelator—turns out to be exactly right.

The value of the math is in showing why that gut feeling works: progress continues underneath you, so any pause there costs less time. It’s a small reminder that mathematics shows up everywhere—even in something as ordinary as tying your shoe on the way to your gate.

Sometimes intuition needs correcting; sometimes it needs confirming, and math is what lets us see the truth either way.

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Imagine you're staring at a puzzle that seems like it should have a solution.

You've got all the right pieces, the numbers add up perfectly, yet something feels off. Today we'll explore a deceptively simple problem that perfectly demonstrates the power of lateral thinking—what mathematicians love to call a "happy idea."

This elegant puzzle comes from British philosopher Max Black, and it's about to show you how the right perspective can turn a seemingly impossible problem into child's play.

The Mutilated Chessboard Problem

Picture a standard 8×8 chessboard with two diagonally opposite corners removed.

Your mission: can you place exactly 31 dominoes (each covering 2x1 rectangles) to perfectly cover every remaining square?

Take a moment to visualize this. You've got 62 squares left to cover, and 31 dominoes that cover exactly 62 squares total. The math checks out perfectly—so this should be possible, right?

The Tempting Trap: Trial and Error

Your first instinct might be to grab some dominoes and start placing them. Go ahead—get a feel for the problem!

You'll quickly notice something frustrating: no matter how cleverly you arrange your pieces, you always seem to hit dead ends. Weird corners that can't be filled. Awkward gaps that frustrate your efforts.

But here's the thing about mathematical puzzles—sometimes the feeling that something's impossible is actually your brain picking up on a hidden pattern.

The question is: can we prove it's impossible without testing every single one of the billions of possible arrangements?

Searching for Restrictions

When facing a "Is it possible to...?" question where your gut says "probably not," mathematicians reach for a powerful tool: restrictions.

Let's check the obvious one first:

• 31 dominoes × 2 squares each = 62 squares covered

• 8×8 chessboard - 2 removed corners = 62 squares to cover

Perfect match! So the area restriction doesn't help us here—the numbers align perfectly.

But wait. There's something else about chessboards we haven't considered...

The Magic Brilliant Idea

Here's where lateral thinking comes in.

The puzzle almost gives us the key by mentioning it's a chessboard—not just any grid, but specifically a chessboard.

Chessboards have colours.

Those alternating black and white squares aren't just a decoration—they're the key to solving this puzzle. Here's the crucial insight:

Every domino, no matter how you place it, must cover exactly one black square and one white square.

Why is this relevant?

Let’s look at our original chessboard with two diagonally opposite corners removed:

It doesn’t matter which two diagonally opposite corners you remove, they’ll both have the same colour.

Let’s say we removed two white corners like in the picture above (the argument works identically if they’re black). We can now find the magic restriction:

• Original chessboard: 32 black squares, 32 white squares.

• After removing two white corners: 32 black squares, 30 white squares.

Now watch what happens as we place dominoes:

• After domino 1: 31 black, 29 white remaining.

• After domino 2: 30 black, 28 white remaining.

• …

• After domino 30: 2 black, 0 white remaining.

It’s impossible to place the 31st domino!

Why? Because every domino must cover one black and one white square, but you've run out of white squares while still having two black squares left uncovered.

The Beauty of Lateral Thinking

This solution is a masterclass in mathematical thinking.

Instead of getting trapped in the overwhelming task of checking every possible arrangement (there are countless billions), we found one simple property—colour parity—that makes the impossibility obvious.

We didn't need to prove the puzzle was impossible case by case, arrangement by arrangement. We found one elegant argument that rules them all out at once. A single, elegant observation led to a completely different angle that made the impossibility crystal clear.

The mutilated chessboard teaches us to trust that "something feels wrong" intuition and then search for the hidden reason why. When you're facing a problem that seems like it should work but doesn't feel right, remember to step sideways and look for the pattern everyone else is missing.

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Your next holidays could solve one of the greatest mysteries in mathematics.

Planning the perfect route through multiple cities seems simple enough—just find the shortest path and you're done. Yet this innocent travel dilemma connects directly to questions that have confused the world's best mathematicians for over half a century. The Clay Mathematics Institute considers it so important that they've placed a million-dollar bounty on anyone who can solve it.

What starts as holiday planning can become a journey into the fundamental limits of human knowledge.

How to Plan Your Holidays

You are planning your holidays to Galicia and you have a list of cities you want to visit: Santiago, A Coruña, Vigo, and Ourense.

You have to start and finish in Santiago, since that’s where you are flying to and from. To make the most out of your holidays, you ask yourself the following question:

What is the best route I can take to visit everything I want?

That is, should I go from Santiago to A Coruña, then to Vigo, then to Ourense and back to Santiago? Or should I go from Santiago to A Coruña but visit Ourense before Vigo? What about going to Vigo first instead of A Coruña?

Fortunately, for your four-city holiday itinerary, it’s quite easy to solve—you can play with Google Maps to find the fastest route.

Algorithms—The Backbone of Computers

The problem above is very common in real-life.

It appears when you plan your holidays, when Amazon decides in which order to deliver their packages, or when you’re running errands and want to optimise your order.

What we just did was follow an algorithm to solve this problem:

1. List out all possible routes you can take.

2. For each route, input it into Google Maps and note the total time.

3. Choose the route with the minimum total time.

An algorithm is a set of rules such that, given any input, they can be followed unambiguously step by step to arrive at a solution to a problem.

If I give you any list of cities, you can run this algorithm and find the best solution.

So, why am I talking about something so simple?

When Simple Becomes Impossible

In the example above we had 3 intermediate cities we wanted to visit: A Coruña, Vigo, and Ourense. There are only 6 possible routes we could take:

1. Santiago—A Coruña—Vigo—Ourense—Santiago

2. Santiago—A Coruña—Ourense—Vigo—Santiago

3. Santiago—Ourense—A Coruña—Vigo—Santiago

4. Santiago—Ourense—Vigo—A Coruña—Santiago

5. Santiago—Vigo—Ourense—A Coruña—Santiago

6. Santiago—Vigo—A Coruña—Ourense—Santiago

You can just try all 6 combinations in Google Maps and find out which is the best route (see the figure above).

Now imagine that instead of 3 intermediate stops you have 4. How many possible orders can you have?

Here's how the math works:

• You first choose which will be your first stop. There are 4 possible choices for this.

• After this choice, you need to choose your second stop. There are 3 possible choices for this (since you already chose your first stop).

• You then choose your third stop, for which you have 2 choices remaining.

• Finally, the last city you visit is the remaining one.

In total, there are 4 x 3 x 2 x 1 = 24 possible routes you can take.

The numbers explode quickly:

• For 5 intermediate stops: 120 routes

• For 6 intermediate stops: 720 routes

• For 10 stops: 3.6 million routes

• For 20 stops: 2.4 quintillion routes (that's 18 zeros!)

For an Amazon driver who might be stopping at 100 different places on a given day, there are so many possible routes that even if all atoms in the universe were computers, and they tried to solve the problem since the Big Bang, they wouldn’t have finished!

The possibilities grow really fast with the number of stops.

This explosion makes the approach described above—listing out all possible routes, computing the time for each, and choosing the fastest one—completely infeasible in real life. Although it works in theory, it doesn't in practice.

P vs NP—The Biggest Problem in Computing

What we've been describing has a name: the Travelling Salesman Problem (TSP).

Imagine a salesman who needs to visit a number of cities exactly once and return to his starting point, while minimizing the total distance traveled. Sounds familiar? That's exactly what we did with our Galician holiday!

The TSP is one of the most important problems in mathematics and computer science.

Above we described an algorithm that works, but it’s very slow. For 4 cities, it needs to check 4 x 3 x 2 x 1 combinations. For 7 cities, this number jumps to 7 x 6 x 5 x 4 x 3 x 2 x 1. In general, for n cities, it needs to check n x (n - 1) x (n - 2) x … x 2 x 1 combinations.

In mathematics, given some natural number n (e.g., 7), the factorial of n, written as n!, is equal to n x (n -1) x … x 2 x 1 (e.g., 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1).

Our algorithm "runs in factorial time". This is just a fancy way of saying that, for n intermediate cities, it needs something like n! operations to solve the problem.

Computer scientists consider an algorithm to be "fast" if it runs in polynomial time: for n intermediate stops, it needs n, n^3 or n^20 operations to solve the problem.

Polynomial growth is much slower than factorial growth:

In other words, any polynomial growth is vastly better than factorial growth.

Here’s the kicker:

Up to this day, we don’t know of any algorithm that solves the TSP in polynomial time.

And here’s something mind-blowing:

If you found an algorithm that solved this problem in polynomial time, you would win one million dollars.

The Million-Dollar Question

In 2000, the Clay Mathematics Institute published a list of seven very important problems in mathematics, which they called the Millennium Problems. Each comes with a $1 million prize for whoever solves it.

Among them is the so-called P versus NP problem, which asks something like the following:

If I can verify whether a solution of a problem is right or wrong in polynomial time, can I also solve the problem in polynomial time?

Problems for which I can verify whether a solution is right or wrong in polynomial time are called NP problems, and problems that I can solve in polynomial time are called P problems. The P versus NP problem asks whether these two classes are the same.

The TSP (with a slight modification) turns out to be an NP-complete problem: if you can solve it in polynomial time, you can solve any other NP problem in polynomial time. Thus, if you found an algorithm to solve the TSP that runs in polynomial time, you could solve any other NP problem in polynomial time, showing that P = NP.

Why This Matters Beyond Holiday Planning

The implications go far beyond vacation planning.

Efficient solutions to the TSP would revolutionize:

• Logistics and delivery: Companies like Amazon, UPS, and FedEx could optimize millions of delivery routes.

• Manufacturing: Circuit board drilling, robotic assembly lines, and supply chain optimization.

• Biology: DNA sequencing and protein folding analysis.

• Cryptography: Many of our security systems rely on problems being hard to solve but easy to verify.

The P vs NP question isn't just a curiosity—it's about understanding the fundamental limits of computation itself.

• If P = NP (meaning every problem that's easy to verify is also easy to solve), it would transform virtually every field that relies on computation.

• If P ≠ NP, it would confirm that some problems are inherently difficult, no matter how clever our algorithms become.

So the next time you're planning a road trip and trying to figure out the best route, remember that you're dealing with one of the deepest unsolved problems in all of mathematics.

And who knows? Maybe your insights over morning coffee could be worth a million dollars…

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In 1901 the whole mathematics community trembled.

Until then, the mathematical world felt secure in its foundations. The rules of logic seemed rigorous and unmistaken, and mathematicians believed they had built an unshakeable system. This changed with a surprisingly simple question by British mathematician and philosopher Bertrand Russell, which shocked the entire mathematics community.

What Russell discovered—now known as Russell’s paradox—revealed a fundamental flaw in the heart of mathematics. Using nothing but strict logical rules, Russell showed how to construct something impossible: an object that has and hasn’t a property at the same time.

But before you start questioning whether 2+2 still equals 4, let me tell you a story about a barber...

Who Shaves the Barber?

Let's start with a puzzle that captures the essence of Russell's puzzling discovery:

The Barber Paradox

The town of Santiago has a barber named Antonio.

Antonio has a very specific job description: he shaves all those men, and only those men, who do not shave themselves.

Now here’s the million-dollar question:

Give this question a try. If you feel like your brain is exploding, that’s exactly how Russell felt when he stumbled upon this logical trap.

Let’s see why this innocent question creates such chaos.

The Building Blocks of Mathematics

To understand Russell’s paradox we first need to take a look at two of the first concepts that any mathematics undergraduate learns: elements and sets.

A set is a collection of things, and such things are called its elements.

The idea is very simple, yet these concepts form the backbone of virtually all modern mathematics.

Let’s see this with some examples:

• The set of all months is the collection formed by January, February… up to December. January belongs to this set; Monday doesn’t.

• The set of even positive numbers includes 2, 4, 6... up to infinity.

• The set of all animals with 4 legs and without trunks includes dogs and horses, but excludes elephants and spiders.

Sets are incredibly flexible—you can collect almost anything into them. And that's precisely where the trouble begins.

The Barber Set

Let's translate our barber puzzle into the language of sets:

Consider the set of all people in Santiago that don’t shave themselves.1

Is Antonio a member of this set?

If he is, Antonio doesn’t shave himself. If he isn’t, he shaves himself.

Let’s work this out:

• Scenario 1: If Antonio belongs to this set, this means he doesn’t shave himself. But wait—Antonio shaves all men who don’t shave themselves. So if Antonio doesn’t shave himself, he must shave himself. Contradiction!

• Scenario 2: If Antonio doesn't belong to this set, then he must shave himself. But Antonio shaves only those men who don't shave themselves. So if he shaves himself, he cannot shave himself. Another contradiction!

We're trapped in an endless loop: assuming Antonio is in the set proves he's not in the set, and assuming he's not in the set proves he is in the set.

It's like discovering that day is night and night is day—pure logical impossibility created from seemingly innocent assumptions.

Sets within Sets: The Plot Thickens

Before we reveal Russell's Paradox, there's one more concept we need: sets can contain other sets.

Consider our set of all months: AM = {January, February, March...}.

Since a set is just a collection, nothing stops us from creating another set that contains AM itself. For example: X = {AM, elephant}.

Here, the entire set of all months AM is an element of the set X, sitting right next to an elephant.

Sets can be elements of other sets—a fact that will prove crucial to Russell's paradox.

Russell’s Paradox: The Earthquake

Now we're ready for the main event. Russell asked a question so simple it seems innocent, yet so profound it broke mathematical certainty:

Consider the set R containing all sets that are not elements of themselves.

In other words: R = {all sets X such that X is not an element of X}.

The master question: is R an element of itself?

• Suppose R is an element of R. Since R is defined as those sets that are not elements of themselves, we conclude that R is not an element of R. Contradiction!

• Suppose R is not an element of R. Since R is defined as those sets that are not elements of themselves, this means that R must be an element of R. Contradiction!

Just like our barber, Russell's set exists in an impossible state—simultaneously belonging and not belonging to itself.

The Solution: When Descriptions Don't Make Sense

So how do we resolve this mind-bending paradox? The answer is surprisingly straightforward, though it took mathematicians some time to fully understand it.

The barber doesn't exist. The description "a barber who shaves all and only those who don't shave themselves" is simply meaningless—it's like describing a "square circle" or a "married bachelor." Just because we can write the words doesn't mean the thing can actually exist.

Similarly, Russell's set R doesn't exist either. The phrase "the set of all sets that don't contain themselves" seems to make sense grammatically, but it's actually not well-defined. It's a description that leads to logical impossibility, which means no such set can exist.

This insight revolutionized how mathematicians think about sets. We learned that not every description automatically corresponds to a real mathematical object. Some descriptions are simply contradictory, even if they sound reasonable at first.

Why This Mattered (And Still Does)

Russell's paradox wasn't just an intellectual curiosity—it was a foundational crisis.

Mathematics had prided itself on logical rigor, yet here was proof that the system contained inherent contradictions. If you can prove both that something is true and false using the same logical rules, how can you trust any mathematical proof?

Real-world implications emerged everywhere:

• Computer science: Early programming languages had to grapple with similar self-referential problems.

• Philosophy: The paradox challenged our understanding of language, meaning, and logical systems.

• Set theory: Mathematicians developed new, more careful axioms (like Zermelo-Fraenkel set theory) to avoid such contradictions.

The Ongoing Legacy

The beauty of Russell's paradox lies not in the problem it created, but in how it forced mathematics to evolve. Rather than breaking mathematics, it made the field stronger by:

1. Forcing precision: Mathematicians became more careful about their foundational assumptions

2. Inspiring new approaches: Modern set theory includes safeguards against self-referential contradictions

3. Highlighting limits: We learned that even logic has boundaries and blind spots.

Today, when programmers debug recursive functions, when philosophers analyze self-referential statements, or when AI researchers grapple with systems that reason about themselves, they're all playing with pieces of Russell's paradox.

In Case You Missed It

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Every time you shop online, check your bank account, or send a private message, you're trusting your most private information to a mathematical idea from the 1970s.

This system, called public key cryptography, solved what back then seemed like an impossible puzzle: how to send secrets to someone you've never met, across networks filled with potential malicious intermediaries, without ever sharing a password beforehand. Before this innovation, secure communication required meeting in person or trusting couriers—imagine if every online purchase required you to fly to the company's headquarters to exchange secret codes. The mathematics behind this system are so elegant that it turned the entire concept of secrecy on its head, making the "lock" public while keeping only the "key" private.

Today, this breakthrough protects trillions of dollars in transactions and billions of private conversations every single day.

The Old Way

Let’s start with the way things were done before the mid-1970s.

For instance, this is the type of cryptography used by the Nazis during WWII with their famous Enigma machine.

Goal: Get a message from a sender to a receiver in such a way that, if intercepted by an unauthorized person in the middle, the unauthorized person can’t understand it.

Idea: Encryption/decryption procedure.

• The sender encrypts the message, making it inaccessible for any unauthorized party. Let’s think of this as putting the message into a box and locking it with a key.

• The receiver decrypts the message, making it readable again. With the same analogy as before, we think of this as opening the box and pulling out the message inside.

Now here is the key difficulty in implementing this:

The sender and the receiver must both have the key that encrypts/decrypts the message (the key that opens/closes the box).

How do we generate the keys in a secure way?

Let’s say the sender is the one generating the keys (i.e., creating the lock and its keys for the box). Then, how does he send the key to the receiver?

We are back at the start! We have a private thing we want to send to a receiver (the key), and we need to do it in a secure way. If we apply the same idea again, we will end up with keys within boxes within boxes within boxes…

Instead, the only way would be to either meet in person to exchange the keys, or have a trusted courier to deliver the key to the receiver. Possible, but very expensive if you want to exchange a lot of messages.

Imagine if every time you wanted to buy something on Amazon, you had to first fly to Seattle to secretly exchange passwords with their security team!

The Great Insight: Public-Key Cryptography

In 1970, British cryptographer James Ellis had an idea so revolutionary it seemed impossible: what if you could send someone a lock, but keep the key?

The idea was the following:

Public Key Cryptography

1. The receiver keeps secret to himself a secret key (also known as private key). This key can decrypt the message. (In our analogy, the receiver creates a special lock and keeps the only key that can open it.)

• The receiver also generates a public key. This is a key that can encrypt the message. (Think about it as a mechanism that can close the lock, but not open it.)

• Since the public key can only encrypt messages, and not decrypt them, there is no need to keep it private. It's like giving away copies of a device that can close your mailbox, but only you have the key to open it and read what's inside.

Think of it like this: you create a special mailbox and put it on a busy street corner. Anyone can drop a letter in (encrypt), but only you have the key to open it and read the contents (decrypt).

One-Way Functions

This idea, as clever as it is, has an essential problem.

The receiver creates a lock and the key to open it (their secret key). Associated with this lock and key, he creates the public key with which senders can encrypt messages for him. The question is:

How can we make sure that, given the public key, it is impossible to recover the private key?

In other words: we need a way to produce a public key from a private key, in such a way that we cannot recover the private key from the public key.

This is what mathematicians call a one-way function: given any input (private key), it produces an output (public key), but it is extremely hard to take an output (public key) and figure out which input (private key) it came from.

Although Ellis himself did not figure out how to do this, his colleague Clifford Cocks implemented such a method a couple of years later.1

RSA—The Cornerstone of Internet Security

The encryption method that Ellis and Cocks came up with underlies most of today’s Internet security.

It is called the RSA cryptosystem, since it was publicly described by R. Rivest, A. Shamir, and L. Adleman in 1977 (Ellis and Cocks’ system was kept classified by the UK government).

The mathematics that underlie the encryption and decryption process, as well as the justification for why they are secure, are unfortunately well beyond the scope of this newsletter (you can watch around min. 10 of Cocks’ talk or check out Wikipedia if you are interested).

However, the way that public and secret key pairs are generated is surprisingly quite simple.

Suppose you are the receiver. You want to generate:

1. A private key: something that only you know.

2. A public key: something related to your private key that everyone can know.

3. These should have the property that, given the public key, it is extremely hard to find out the private key.

The RSA cryptosystem uses prime numbers: numbers that can only be divided by themselves and 1—like 2, 3, 5, 7 or 11, but not 4 (it can be divided by 2). It works as follows:

1. The private key will be a very large prime number p, say 7,879.

2. To generate the public key, we first choose another very large prime number, say 6947. The public key will then be the product N = p x q = 7,879 x 6,947 = 54,735,413.

Now here is the crucial thing:

Given a very large number like 54,735,413, there is often no better way to find its factors than pure guessing.

That is, if I only give you 54,735,413 (the public key), to find a number that divides it (that is, 7879, the private key), you have to try, one by one, all the numbers from 1 up to 54,735,413.

Technically you can do it, but for truly 'very large' numbers (see below), you will die before you guess it!

Very Large Really Means Very Large

The primes I chose, 7,879 and 6,947, are by no means 'very large' primes.

When I say very large, I really mean very large, something like:

4369518557955514614435635868603803074368490343736...548163288969925913941393434090233759524802729.

These numbers can have more than 300 digits!

The numbers I chose were only for clear exposition (and to be able to write them down!).

To put this in perspective: if you tried to guess one of these 300-digit prime factors by checking one billion numbers every second, it would take you longer than the age of the universe to find it!

Where You Use This Every Day

You might not realize it, but you rely on RSA encryption dozens of times each day:

• Online Shopping: Every time you enter your credit card information, RSA protects it from hackers

• Banking: Your online banking sessions are secured using these mathematical principles

• Messaging: Apps like WhatsApp use modern versions of public key cryptography to protect your conversations

• Email: Gmail, Outlook, and other email services encrypt your messages in transit

• Software Updates: Even downloading apps or system updates relies on these keys to verify authenticity

The Future Challenge

There's a catch to this seemingly perfect system: quantum computers.

While today's computers would take billions of years to crack RSA encryption, a sufficiently powerful quantum computer could potentially do it in hours or days. That's why cryptographers are already working on "quantum-resistant" encryption methods—but that's a story for another newsletter!

The Beauty of Cryptography

The next time you buy something online or check your bank account, remember that you're witnessing one of mathematics' most elegant solutions to an ancient problem. What started as a theoretical breakthrough in the 1970s now protects trillions of dollars and billions of private conversations every day.

The beauty of public key cryptography lies in its simplicity: anyone can lock the box, but only you can open it. It's a mathematical trick that turned the impossible into the everyday, proving once again that sometimes the most profound innovations come from the most elegant ideas.

And the best part? The math that makes it work is literally public—yet remains unbreakable by design.

In Case You Missed It

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Your brain is a terrible probability calculator.

What feels obviously true about randomness and coincidence is often wrong, leading us to make poor decisions in everything from gambling to cybersecurity. Mathematicians have known this for decades—the gap between human intuition and mathematical reality creates some of the most shocking discoveries in probability theory. Even the smartest people consistently fail at estimating the likelihood of seemingly rare events, costing them money, opportunities, and credibility.

Today's simple birthday question will prove just how unreliable your instincts really are.

The Birthday Problem

Imagine tonight Barça and Madrid clash in El Clásico.

Picture this: 22 players sprint across the field (11 per team), plus one referee keeping order. That's 23 people total. Now, here's a bet that I offer you:

If any two people on that field share the same birthday, I'll give you €100. If not, you give me €100.

When Math Beats Intuition

If I offered you €100 for heads on a coin flip but asked for €100 on tails, you'd know it's a fair deal—50/50 odds.

Instead, I am offering you a deal where you win 100€ if there is a common birthday, and you lose 100€ if not. Therefore, the question you should ask yourself is:

Is it more likely that there is a common birthday, or that there isn’t?

More specifically:

What is the probability that there is a common birthday?

Your intuition probably says birthdays are spread out randomly, so finding a match among just 23 people seems unlikely. Is it really?

Probability—a powerful branch of mathematics—helps us make optimal decisions when facing uncertainty.

The Hard Way vs. The Smart Way

Assuming we don’t know the birthdays of any of the players or the referee, how can we tackle this problem?

The Hard Way

For two people to have the same birthday, we could calculate the probability of every possible scenario:

• There are exactly 2 people among the 23 with the same birthday (that is, among the rest of the people none of them share the same birthday).

• There are exactly 3 people with the same birthday.

• There are exactly 4 players with the same birthday.

• And so on and so forth, until the (pretty much impossible) case that everyone has the same birthday!

We could try to compute the probability of all of the scenarios above, and then add all of them together. However, these are a lot of cases to consider!

Can we do any better?

The Smart Way: Your Favourite Probability Trick

I’m going to tell you one of the most useful tricks to deal with probability computations that seem to have too many cases.

It’s based on the following simple observation:

Exactly one of the following things must be true:

1. At least two people share a birthday.

2. No one shares a birthday.

Since one of the two things above must happen, their probabilities add up to 1 (or 100%).

If we can find the probability that no one shares a birthday, we can subtract from 1 to get our answer. This turns a complex problem into a manageable one.

The Calculation

(Feel free to skip this section if you prefer the bottom line!)

Imagine an unmarked calendar with all 365 days available.

• Person 1: Can be born on any day. Probability of no conflict = 365/365 = 1.

• Person 2: Can be born on any of the remaining 364 days. Probability = 364/365.

• Person 3: Can be born on any of the remaining 363 days. Probability = 363/365.

• Person 4: 362/365.

• ...and so on for all 23 people.

The probability that NO ONE shares a birthday is:

The Shocking Answer

The probability that at least two people share a birthday is approximately 51%!

You should absolutely take my bet—you're slightly more likely to win than lose.

Even more surprising: 23 is the magic number where the odds tip in your favor. With 22 people (no referee), the probability drops just below 50%, making the bet unfavorable.

The numbers get dramatic quickly:

• 30 people: 70% chance of a match.

• 50 people: 97% chance.

• 57+ people: >99% chance.

Why Our Intuition Fails Us

The secret lies in pairs.

Among 23 people, there aren't just 23 potential birthdays to consider—there are 253 possible pairs of people who could share a birthday:

• Person 1 could match with any of 22 others.

• Person 2 could match with any of the remaining 21 (we already counted Person 1 matching Person 2 above).

• Person 3 with any of the remaining 20.

• ...and so on.

With 57 people, you get 1,596 possible pairs!

Each pair represents another chance for a birthday match, which is why the probability climbs so rapidly.

Beyond the Football Field

This principle appears everywhere:

• Cybersecurity: the birthday paradox limits how secure hash functions are against collisions, enabling “birthday attacks” that exploit this math to find duplicates faster than brute force.

• DNA evidence: the birthday problem helps assess the chance of accidental matches within large DNA databases, critical for understanding false positive risks and non-uniqueness of DNA profiles.

• Social networks: the birthday problem explains why in relatively small groups you often find mutual friends or shared connections, reflecting how quickly overlaps appear in large networks

The Takeaway

Next time someone offers you a birthday bet with 23+ people, remember: mathematics beats intuition. The birthday problem reveals how our brains struggle with probability and combinatorial questions.

So, would you take my €100 bet now?

What other counterintuitive probability problems fascinate you? Reply and let me know what you'd like to explore next!

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Have you ever wondered why heavy cargo ships float, yet a tiny 1kg rock sinks deep into the ocean? Why does an underwater air bubble rise up towards the surface, instead of staying still or sinking further down? Why do we humans float in the ocean?

The famous Greek mathematician, physicist, and inventor Archimedes of Syracuse wondered about the same questions while taking a bath over 2,000 years ago. What he discovered is one of the most important laws of fluid mechanics—and it is much simpler than you think.

Archimedes’ Take on Buoyancy

Archimedes’ principle is one of those physical laws that, despite being very simple, has far-reaching applications. In his treaty On Floating Bodies, the great Archimedes proposed the following principle:

Any object, totally or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

Let’s unpack what this actually means.

When Things Go Down

As shown in the figure, suppose we have a 5kg piece of iron (left). We submerge it into a tank full of water, which causes some of the water to be displaced—in our case, 2kg of water (right). Archimedes’ principle says that the iron piece will be buoyed up by a force equal to those 2kg. So, after submerging the iron piece:

• Its weight of 5kg pulls it down.

• The buoyant force pulls it up with a force of 2kg.

• Overall, there is a downward pull of 3kg (see scale on right).

In this case, the iron piece still has a net downward force after submerging it, so it will sink down.

When Things Rise Up

Now here comes the key insight:

The amount of water that is displaced depends on the size of the object, not on its weight.

Why is this game-changing?

Suppose that, instead of having a 5kg piece of iron, we have a 5kg piece of wood. Wood is way less dense than iron (it’s less 'compact'), which means that 5kg of wood occupies a much larger volume than 5kg of iron. If we crunch the numbers, we can find that it will displace something like 10kg of water (depending on the type of wood).

So, what happens?

• The wood’s weight of 5kg pulls it down.

• The buoyant force pulls it up with a force equal to the water it displaces, i.e., 10kg (Archimedes’ principle).

• Overall, there is an upward pull of 5kg.

Since the net force points upward, the wood piece will float up towards the surface.

The Basketball Experiment You Can Try Today

This upward pull is a common phenomenon in real-life.

Have you ever tried fully submerging a basketball into the ocean? You can certainly do it, but you will feel how it pushes up towards the surface. In fact, in this extreme case it does it so strongly that it will jump high into the air if you release it!

As shown in the image, if you instead leave the basketball on the surface of the water, you’ll find that it’s only partially submerged: most of it is above the surface, while a small portion is submerged below the surface.

Archimedes’ principle explains this. For the ball to be in equilibrium—neither sinking down nor going up, just floating steadily—the downward pull must be exactly equal to the upward pull.

• The downward pull is constant and equal to the weight of the ball.

• The upward pull depends on the amount of water displaced (Archimedes’ principle).

Physics tells us that the ball will submerge precisely enough so that the amount of water it displaces is exactly equal to its own weight.

This is why a cork is nearly fully above water when floating (it needs to displace very little water), yet a piece of wood might be half-submerged.

Why Do Massive Ships Float?

Cargo ships are mind-bogglingly heavy.

They are not only made of heavy steel, but also carry massive amounts of equipment, containers, people, food, fuel and water. The largest bulk carriers can weigh up to 500,000,000 kg (that’s roughly 100,000 elephants!).

How can these floating giants stay afloat?

Archimedes’ principle gives the answer: by displacing an enormous amount of water.

If you looked underwater, you’d discover that a ship has a considerable portion of its total size submerged under water. It’s 'partially submerged', as we discussed above. The amount of water it displaces is precisely enough to keep itself floating.

When cargo ships are designed, engineers estimate their maximum weight as the sum of the weights of the cargo, hull, fittings, equipment, propelling and auxiliary machinery, piping systems, electrical and electronic gear, fuel, water, consumable stores, passengers and crew (plus a margin of a few percent for weights that are underestimated).1 They then design the shape of the ship such that, given this total maximum weight, the ship is big enough to displace enough water to float.

Thanks to a mathematician's bathtub eureka moment 2,000 years ago, we now have container ships carrying your Amazon packages across the Pacific!

My Scuba Diving Adventure (and How Physics Saved the Day)

I’ve just come back from a trip in Southeast Asia where I learned to scuba dive. In fact, it was this experience that motivated me to write this post.

Scuba divers use weight belts (usually made of lead pieces) to sink underwater. When they reach the desired depth for the dive—that is, they don’t want to sink down any further, nor rise up towards the surface—they use Archimedes’ principle:

By inflating their BCD (buoyancy control device, a vest that can be inflated) with just the right amount of air such that the volume of water this air displaces is equal to the weight of the scuba diver plus their weight belt, they achieve neutral buoyancy and hover motionless underwater.

If a scuba diver wants to ascend slightly to avoid a rock, or descend to examine something closer, they could inflate or deflate their BCD to achieve it.

But here’s the really cool part!

In reality, scuba divers use their own lungs to move up and down once they have reached neutral buoyancy. Our lungs expand and contract as we breathe, thus displacing more or less water. By adjusting their breathing patterns—taking deeper breaths if they want to rise up, or exhaling more fully if they want to sink down—scuba divers can rise and sink as they wish.

Try This Yourself

Next time you go swimming, try it yourself! If you are floating in the water and take a deep breath, what do you think you will feel?

This is Archimedes’ principle in action, powered by your own lungs!

More Amazing Applications of Archimedes’ Principle

Hot Air Balloons: the heated air inside is less dense than the cool air outside, so the balloon displaces more weight in air than it actually weighs—up it goes!

Submarines: They control their buoyancy by flooding or emptying ballast tanks with water, changing how much water they displace.

Ice Floating: Ever wonder why icebergs are mostly underwater? Ice is less dense than water, but only slightly—so about 90% stays submerged while 10% peeks above the surface.

Hydrometers: These devices measure liquid density by seeing how deep they sink—the denser the liquid, the higher they float!

In Case You Missed It

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Have you ever made two completely opposite decisions about the same problem—simply because it was presented differently? Don't worry, you're not alone. In fact, even experts fall into this trap regularly.

The way information is presented to us has a profound effect on how we process it and act on it. This phenomenon, called the "framing effect," reveals something puzzling about human decision-making: our choices often depend less on facts and more on how those facts are framed.

The Asian Disease Problem: A Classic Experiment

Consider this fascinating decision-making scenario, designed by psychologists Kahneman and Tversky to study how framing affects our choices:

Scenario 1
Imagine that the United States is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows:

• If Program A is adopted, 200 people will be saved.

• If Program B is adopted, there is a one-third probability that 600 people will be saved and a two-thirds probability that no people will be saved.

Now consider this alternative scenario:

Scenario 2
Imagine that the United States is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume that the exact scientific estimates of the consequences of the programs are as follows:

• If Program C is adopted, 400 people will die.

• If Program D is adopted, there is a one-third probability that nobody will die and a two-thirds probability that 600 people will die.

Take a moment to decide which option you'd choose in each scenario before reading on.

Inconsistency of Choices

If you're like most people, you chose Program A in the first scenario and Program D in the second.

But here's the shocking truth: these choices are logically inconsistent!

Why? Because the programs in both scenarios are identical—just described differently:

• Program A and Program C are the same outcome: 200 people live, 400 die

• Program B and Program D are the same outcome: 1/3 chance everyone lives, 2/3 chance everyone dies

So why do we make different choices? It all comes down to framing.

How Framing Shapes Our Decisions

Our choices are dependent on the way information is presented to us.

In the first scenario, the options are presented as "gains" or "wins":

• We start with a mental baseline of everyone dying

• Program A offers a certain gain (200 people saved)

• Program B offers a risky gamble (potentially save everyone or save no one)

When dealing with gains, we tend to be risk-averse—we prefer the sure thing.

In the second scenario, the same options are presented as "losses":

• We start with a mental baseline of everyone living

• Program C guarantees a loss (400 people die)

• Program D offers a chance to avoid any loss (though with risk)

When facing losses, we become risk-seeking—we'll gamble to avoid a sure loss.

If presented as a win we go for the sure thing; if presented as a loss we gamble. As Kahneman puts it:

"Risk-averse and risk-seeking preferences are not reality-bound. Preferences between the same objective outcomes reverse with different formulations."

Real-Life Threats: Experts Also Get It Wrong

This isn't just an academic exercise. When researchers presented these same scenarios to public health professionals—the people who make decisions about vaccines and other health policies—they fell into the same trap.

Let that sink in: the experts who determine public health strategy can be influenced by simple changes in how options are presented, not by the actual outcomes.

Framing and Moral Judgments: Helping the Poor while Favouring the Rich

The following is another great example of the inconsistency caused by framing effects. I now realise that many of my political and economical ideals might have been completely illogical!

Scenario A
A government wants to promote families having more children. To do so, families with more children will pay less taxes: there is a baseline tax for having no children, and for each additional child, a family will receive a tax reduction.

Most people will be opposed to this idea. The idea of the favouring the rich with a larger reduction is completely unacceptable.

Now consider this alternative:

Scenario B
A government wants to promote families having more children. To do so, families with more children will pay less taxes: there is a baseline tax for having two children, and having fewer than this will result in a tax increase.

Here, most people are again opposed to this idea: penalising the poor more than the rich is completely unacceptable.

But guess what: these preferences are logically contradictory!

Why Our Moral Intuitions Fail Us

Let's work through a concrete example:

Imagine two families with no children—one rich (Family R0) and one poor (Family P0). Both pay taxes of 20%.

Now consider two additional families, each with two children—one rich (Family R2) and one poor (Family P2).

The government’s aim to promote families having more children translates into Family R2 paying less taxes than Family R0, and similarly Family P2 paying less taxes than Family P0.

According to our response to the first scenario, we believe the tax reduction for Family P2 (compared to P0) should be at least as large as the reduction for Family R2 (compared to R0). For example:

• Rich families: R0 pays 20%, R2 pays 18% (2% reduction)

• Poor families: P0 pays 20%, P2 pays 17% (3% reduction)

Now consider this instead from the point of view of the 2-children families:

• Rich families: R2 pays 18%, R0 pays 20% (2% increase)

• Poor families: P2 pays 17%, P0 pays 20% (3% increase)

If we want to favor the poor more than the rich, we necessarily (want to) penalise them more than the rich!

Rethinking Our Moral Judgments

Our intuitive, fast-responding System 1 (as Kahneman calls it) delivers an immediate response to questions about fairness: "when in doubt, favor the poor."

Yet, examples like the one above show that this simple moral rule does not produce logical results, generating contradictory answers to the same problem. As Kahneman puts it, and as sad as it may sound:

"Your moral feelings are attached to frames, to descriptions of reality rather than to reality itself."

What Do You Think?

Now that you understand how framing affects your judgments:

Share your thoughts in the comments! Have you ever noticed yourself making inconsistent choices because of framing?

In Case You Missed It

• The Limits of Democracy: What Arrow’s Impossibility Theorem Reveals about Fair Elections

• How You Can Make Better Choices (Spoiler: Self-Interest Fails)

• Infinity and Beyond: How Some Infinities Are Bigger Than Others

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What if I told you that the most significant events in your life and in world history were, by definition, impossible to predict?

That's the fascinating premise of Nassim Nicholas Taleb's 'The Black Swan', the most thought-provoking book I've read this year.

In a world plagued with forecasting, planning, and predicting, I believe Taleb's work is a must-read. Through a combination of philosophy, statistics, and psychology, he reveals why our understanding of randomness is fundamentally flawed—and how this misunderstanding shapes everything from financial markets to personal success.

What is a Black Swan?

The central idea of the book (as its title suggests) is that of a Black Swan. Taleb defines a Black Swan as an event with three key attributes:

"First, it is an outlier, as it lies outside the realm of regular expectations, because nothing in the past can convincingly point to its possibility. Second it carries an extreme impact […]. Third, in spite of its outlier status, human nature makes us concept explanations for its occurrence after the fact, making it explainable and predictable."

The financial crisis of 2008, Hitler’s rise to power and the subsequent world war in the 1930’s, the collapse of the Soviet Union in the 90’s, the rise of the Internet, the COVID-19 pandemic…they are all examples of Black Swans: unexpected, extremely consequential and explainable a posteriori.

Note that, in retrospect, we now think that all these events were somehow predictable—that it was only a matter of time before they happened. This is precisely the third property of a Black Swan: we now think it was explainable and predictable (see the narrative fallacy below), but it certainly was not before it happened.

Think about it: if the 2008 financial crisis was truly predictable, would investors have continued inflating the bubble right until it burst? Are we really smarter today, or are we just benefiting from hindsight?

Why We Miss Black Swans

The Error of Confirmation

We naturally seek evidence that confirms our existing beliefs rather than looking for contradicting evidence: this is the error of confirmation.

However, Taleb argues that we can get much closer to the truth by finding negative instances than positive ones—also known as a negative empiricist approach.

Consider this example: if you believe your friend John is not a criminal, watching him commit no crime for years doesn't prove your belief. All you can say is that there is no evidence that he is a criminal.

On the other hand, a single criminal act immediately disproves your belief—now there is evidence that he is not lawful.

This asymmetry has profound implications: we know what is wrong with much more confidence than we know what is right.

Yet, we keep looking for evidence of what we believe is right!

Absence of evidence (observing cases that agree with our hypothesis) is very different from evidence of absence (observing a single case that contradicts our hypothesis).

The Narrative Fallacy

Humans are storytelling creatures. We instinctively create logical narratives to explain random sequences of events, making the world appear simpler and more predictable than it actually is.

We’ve talked about something similar when discussing the Linda Problem. There, we showed that adding more details to a story—making it more causally explainable—lead us to believe it was more likely, despite that being mathematically impossible.

This inherent bias to create stories is what Taleb and others call the narrative fallacy.

When we read a successful person's biography, we create a coherent story of how their talent and decisions led inevitably to their success. This satisfies our need for causality, but dramatically underestimates the (huge) role of chance (more on this below).

The Problem of Silent Evidence

History shows us only the winners—the successful writers, philosophers, and entrepreneurs who made it to the top. What we don't see are the thousands of equally talented individuals who failed along the way.

Taleb argues that, while someone like Gabriel García Márquez was certainly brilliant, his Nobel Prize wasn't just about skill—luck played a crucial role. This is hard for us to accept because we want to believe success comes from merit alone.1

The math is simple and reveals our flawed understanding of randomness. The probability that any specific writer will achieve García Márquez's success is tiny. But with hundreds of thousands of writers working at any given time, the probability that someone will get extraordinarily lucky becomes quite high.

It just happened to be Márquez who won the luck lottery!

I find this idea fascinating—luck is so important in our lifes. I recommend watching this great video by Veritasium if you are still not convinced (Spanish version here).

The reference point argument: do not compute odds from the vantage point of the successful writer/businessman/intellectual, but from all those who started in the cohort.

Two Types of Randomness

Taleb divides randomness into two domains with fundamentally different properties: Mediocristan and Extremistan.

Mediocristan: Where Outliers Don't Matter

In Mediocristan, extreme outliers have minimal impact on the collective.

Imagine measuring the heights of everyone in a football stadium. If you suddenly added the tallest person on Earth to your sample, it would barely change the average—not even a 1cm difference!

Most "physical" attributes belong to Mediocristan: height, weight, caloric consumption per day, life expectancy…

Statistical tools work reliably in Mediocristan because extreme outliers won't skew your results.

Extremistan: Where Single Events Change Everything

In Extremistan, a single outlier can completely dominate the aggregate.

Using the same football stadium example but measuring wealth instead of height: if you added the richest person in the world to the crowd, the average wealth would skyrocket. One outlier completely breaks your statistical model.

Most "social" variables belong to Extremistan: wealth distribution, book sales per author, social media followers, number of speakers per language…

The Essential Problem

We act as we live in Mediocristan, yet many (most) relevant variables in today’s world are indeed part of Extremistan.

We act as outliers are the exception to the norm and they can be safely ignored, yet they are so consequential that they drive the course of history.

Taleb’s Central Thesis: We Simply Cannot Predict

Our Overconfidence Problem

First, there is what Taleb calls “epistemic arrogance”.

This refers to an overconfidence that we have in our knowledge. It has been demonstrated over and over again in experiments: some experiments show that, on average, people are 22 times more confident in their knowledge than they should be!

Note that this does not refer to knowing or not knowing, but rather to being aware of what one does not know.

Paradoxically, more information often makes this worse. The more data we consume, the more patterns we think we see, and the more confident we become in our flawed predictions.

We see random noise and mistake it for information.

The Unpredictability of History

History doesn't evolve gradually but jumps suddenly through Black Swan events. As Taleb puts it, "history does not crawl, it jumps."

Since Black Swans are, by definition, unpredictable and extremely consequential, this means history itself is fundamentally unpredictable.

The Way Forward: Skeptical Empiricism

Rather than false certainty, Taleb advocates for "skeptical empiricism":

• Skepticism: Question everything, especially narratives that "make sense".

• Empiricism: Prioritize practical experience over elegant theory.

Just because something "makes sense" doesn't mean it works, and something might work perfectly without having a neat explanation!

Taking Advantage of Black Swans

"Put yourself in situations where favourable consequences are much larger than unfavourable ones."

Rather than wasting your time trying to predict Black Swans (if you could predict it, it wouldn’t be a Black Swan), position yourself to benefit from them. This means:

1. Maximize exposure to positive Black Swans: Seize opportunities that, while unlikely to succeed, would have enormous benefits if they did. This is not an easy task, as you'll need to face many small losses for a long time. However, a single positive Black Swan will compensate all the slow bleeding—in Extremistan, outliers dominate the aggregate.

   “Size any opportunity, or anything that looks like an opportunity.”

2. Minimize exposure to negative Black Swans: Avoid situations that seem safe but could be catastrophic in rare circumstances.

I think this is very insightful. We have a tendency to the opposite: we like ventures that seem not too risky, that sit in the middle of the risk spectrum. Instead, Taleb advocates for a hyperagressive and hyperconservative mentality. keeping the bulk of your investments or actions in the hyperconservative part to limit the effect of a negative Black Swan, yet being hyperagressive in the remaining to get an exposure to the positive Black Swan.

Here's Taleb's financial example:

Instead of putting your money in "medium risk" investments (how do you know it is medium risk? […]), you need to put a portion, say 85 to 90 percent, in extremely safe instruments, like Treasury bills […]. The remaining 10 to 15 percent you put in extremely speculative bets […]. That way […] no Black Swan can hurt you at all, beyond your 'floor', the nest egg you have in maximally safe investments. […] You are "clipping" your incomputable risk, the one that is harmful to you.

This barbell strategy—extremely safe on one end, extremely risky on the other—creates a profile that's resilient to negative Black Swans (if one comes, you lose at most 10/15% of your capital) while has a positive exposure to the Black Swan (if one comes, you can make enormous profit).

Conclusion

The Black Swan offers a profound lesson in intellectual humility. It challenges us not just about what we know, but about what we think we know and what we can know.

History has shown many times how terrible we are at prediction—yet we continue making forecasts with extreme confidence. This overconfidence isn't only a philosophical problem; it has real consequences in our financial systems, public policies, and personal lives.

The inherent unpredictability of our world isn't something to fear but to embrace strategically. As Taleb suggests, we can position ourselves to benefit from positive Black Swans while limiting our exposure to devastating ones. As Taleb puts it:

Knowing that you cannot predict doesn’t mean you cannot benefit from unpredictability—N. Taleb.

In Case You Missed It

• Infinity and Beyond: How Some Infinities Are Bigger Than Others

• The Smartest Way to Bet: Understanding Kelly Criterion

• Infinitely Close or Perfectly Equal? Why 0.999... = 1

• Two Selves: Living for the Moment or the Memory?

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In 1971, physicist Joseph C. Hafele and astronomer Richard E. Keating took four atomic clocks—some of the most precise clocks ever created—and flew them around the world on commercial airliners.

What they found upon returning was mind-blowing. The flying clocks showed a different time than identical atomic clocks that had remained in the lab. The difference was tiny (billionths of a second) but unmistakable.

Far from being shocked, Hafele and Keating were delighted. This bizarre result confirmed exactly what Einstein's theory of relativity had predicted over 50 years earlier: time itself flows differently depending on how you're moving.

Time dilation, length contraction, relativity of simultaneity…these concepts might sound like science fiction, but they're the fundamental reality of our universe.

Two Flavours of Relativity: Special and General

Einstein's theory of relativity—arguably the most revolutionary scientific advancement of the 20th century—comes in two distinct "flavors".

• Special relativity deals with objects moving at constant velocity—no speeding up, slowing down, or changing direction. It's "special" because it only applies to this particular (and simplified) situation.

• General relativity expands these principles to all types of motion, including acceleration and the effects of gravity. It's our most precise description of how space, time, and gravity work on large scales.

Today we'll focus on special relativity—the simpler theory that still manages to completely transform our understanding of reality.

Two Simple Ideas That Changed Everything

One of the most beautiful aspects of special relativity is its elegant simplicity.

The entire theory—with all its mind-bending consequences—follows from just two basic principles that Einstein proposed in his revolutionary 1905 paper "Zur Elektrodynamik bewegter Körper" (“On the Electrodynamics of Moving Bodies”):

Postulates of Special Relativity

1. The Principle of Relativity: the laws of physics work exactly the same way in all frames of reference moving at constant speed.

2. The Principle of Light Constancy: light always travels at the same speed (roughly 300,000 km/s) for all observers, regardless of how the light source or the observer is moving.

These seemingly innocent principles lead to profound consequences that completely reshape our understanding of reality.

Physics Doesn't Care How Fast You're Going

Imagine waking up in a perfectly smooth-riding train with all windows covered. Could you tell whether you're moving or standing still?

Try walking down the aisle—it feels normal. Toss an apple to a friend—it travels in a straight line. Drop your phone (accidentally, of course)—it falls straight down. No physical experiment you could perform inside this sealed train would tell you whether you're travelling across the countryside or sitting motionless at the station.

This principle—that the laws of physics work identically in any reference frame moving at constant velocity—was first articulated by Galileo Galilei nearly 300 years before Einstein. What makes Einstein's contribution revolutionary is what comes next.

The Speed of Light's Strange Behaviour

Now we reach the truly revolutionary part of Einstein's theory.

Let's return to our train example. If you're on a train moving at 100 km/h and you throw a ball forward at 50 km/h:

• From your perspective on the train, the ball moves away at 50 km/h.

• To someone standing on the platform watching, the ball travels at 150 km/h (train speed + throw speed).

This matches our everyday experience—velocities add up. But Einstein proposed something radical: light doesn't play by these rules.

If you turn on a flashlight while on that moving train:

• From your perspective, the light beam travels at exactly 300,000 km/s.

• From the perspective of the person standing on the platform, the light also travels at exactly 300,000 km/s—not 300,100 km/s as we might expect.

This strange behaviour—that light's speed remains constant regardless of the motion of its source or observer—has been confirmed by numerous experiments, beginning with the famous Michelson-Morley experiment in 1887.

This seemingly innocent fact forces us to reconsider the very nature of space and time themselves.

The Mind-Bending Consequences of Special Relativity

From Einstein's two simple principles emerge consequences so profound they completely reshape our understanding of the universe:

Time Dilation: Your Clock Isn't My Clock

Remember the flying atomic clocks I mentioned at the beginning? Here's what's happening: time itself passes more slowly for moving objects. This isn't a clock error or measurement mistake—it's actual time flowing at different rates.

The faster you move, the more time slows down relative to "stationary" observers. If you could travel at 99% the speed of light, you would experience minutes passing while years elapse for those you left behind.

This effect happens even at everyday speeds, just to an incredibly tiny degree. The GPS satellites orbiting Earth move fast enough that they experience time about 7 microseconds slower each day than we do on Earth's surface. Without accounting for this effect, your navigation apps would accumulate errors of about 2 kilometers per day!

Time dilation is the phenomenon by which the time measured by a moving observer is larger (it is 'dilated') than the one measured by a stationary observer.1

Time is slowed down for a moving observer.

Length Contraction: Objects Shrink When They Move

Just as time stretches with motion, space contracts. A moving object appears shortened in the direction of its motion.

This leads to mind-bending scenarios: a 20-meter spacecraft traveling at extremely high speeds might appear just 10 meters long to a stationary observer. Theoretically, a spacecraft that wouldn't fit inside a tunnel while stationary might fit perfectly when zooming through at near-light speed!

Like time dilation, this effect is real but negligible at everyday speeds. You won't notice your car getting shorter during your commute.

Length contraction is the phenomenon by which the distance measured by a moving observer is shorter (it is 'contracted') than the one measured by a stationary observer.

Distances are shortened for a moving observer.

Relativity of Simultaneity: "At The Same Time" Loses Meaning

Perhaps most profound of all: events that appear simultaneous to one observer may occur at different times according to another observer in a different reference frame.

Imagine lightning striking both ends of a moving train. A person standing on the platform might see both strikes happen simultaneously. But a passenger on the train would see the lightning strike at the front of the train first, followed by the strike at the back.

Relativity of Simultaneity: the simultaneity of two events is relative.

Whether two events happen at the same time or not depends on the observer, and both are right within their own reality.

Why This Matters

Special relativity might seem abstract, but its effects touch our lives in surprising ways.

Every time you use GPS navigation, you're benefiting from relativistic corrections. The atomic clocks on GPS satellites experience both time dilation (from their speed) and gravitational time differences (from being farther from Earth's mass). Without accounting for these effects, location errors would accumulate at a rate of about 10 kilometers per day, rendering the system useless.

Modern medicine and scientific research rely on particle accelerators, where subatomic particles are accelerated to near light speed. Engineers must account for relativistic effects when designing these machines—the particles become measurably heavier as they approach light speed due to the famous E=mc² relationship.

Beyond this and many other practical applications, Einstein's theories fundamentally changed how we understand reality. Time isn’t absolute. Space isn’t universal. Cause and effect are constrained by the speed of light.

Our intuitive separation of space and time is merely an approximation that works at the slow speeds of human experience.

A New View of Reality

At human speeds—walking, driving, even flying—the effects of relativity are so tiny that our everyday experience feels consistent with Newton's classical view of fixed space and time. It's easy to believe these are fundamental, unchanging parts of reality.

But special relativity reveals a different truth: time slows, lengths contract, and simultaneity shifts depending on how you're moving. These aren't perceptual illusions or measurement errors—they reflect the actual structure of our universe.

The universe is stranger and more wonderful than common sense suggests. As you go about your day, remember that the time on your watch, the distance to your destination, and even what events are happening "right now" across town—all depend on how you're moving through our relativistic universe.

What questions do you have about relativity? Reply to this email or post a comment and let me know what other concepts you'd like me to explain in upcoming issues!

In Case You Missed It

• The Hidden Psychology of Change: How We Value Gains and Losses

• Inside ChatGPT: How Large Language Models Predict the Future of Text

• Financial Derivatives II–Options

• The Impossible Race: How a Tortoise Challenged Mathematics for 2,500 Years

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Infinity is a mind-blowing mathematical concept that challenges our everyday intuition.

Remember The Infinite Hotel thought experiment? It revealed some truly surprising results:

• The Infinite Hotel, even if completely full, can fit one more person.

• The Infinite Hotel, even if completely full, can fit infinite more people.

• The Infinite Hotel, even if completely full, can fit infinite groups of infinite people

Today we consider the following question: is there anything that The Infinite Hotel simply cannot fit?

How Mathematicians Measure Things

Comparing the size of two real-life hotels is straightforward.

If one hotel has 30 rooms and another has 200 rooms, one immediately knows that the second one is larger. Why? Because 200 is larger than 30? And why is this? Because 200 is 30 plus something else.

Why I am stating these obvious things? Because, once again, our intuitive "obvious" reasoning fails when dealing with infinity.

It is not necessarily true that infinite plus something is larger than infinity. Remember The Infinite Hotel: it can fit infinite people and be full, and also infinite people plus one more and still be full. The hotel hasn’t changed, so infinity is as big as infinity plus one.

What about infinite things?

We need a different way to compare the sizes of two infinite things.

Let me use the idea of a hotel size to introduce the relevant concept:

Comparing Hotels

We say that two (infinite) hotels have the same size if there is a way to match every room in one hotel with one room in the other, in such a way that all rooms are covered exactly once.

Let’s see some examples to understand this better:

Example 1: Same Finite Size

Let’s take two hotels with 4 rooms each. They have the same size in the obvious sense, and also with our new definition—here is a pairing:

This is a valid matching to compare sizes, since:

1. Every room in Hotel A has an assigned room in Hotel B.

2. No two rooms in Hotel A are assigned the same room in Hotel B.

3. Every room in Hotel B has a match in Hotel A.

Example 2: Different Finite Sizes

Now let’s take two hotels, one with 4 and the other with 5 rooms. Here is what happens when you try to make a pairing between them:

The attempt in the left is not a valid pairing because the last room in Hotel 2 is not matched by any room in Hotel 1.

The attempt in the right is also not a valid pairing because the last two rooms in Hotel 1 are assigned the same room in Hotel 2.

No matter how hard you try, you won’t be able to come up with a valid pairing for these two hotels.

The Magic of Infinite Assignments

Now let’s look jump to infinity and see some more interesting matchings. For example, let’s see that a hotel with infinite rooms is as large as one with infinite rooms plus one extra room.

I’ll number the rooms of the first hotel by 1, 2 , 3… up to infinity. For the second hotel, let’s number its extra room as 0. The following pairing shows that these two hotels have the same size:

With this clever shifting of room numbers, we've created a perfect one-to-one matching. Therefore, these two hotels have the same size.

Similarly, being a bit ingenious, you can show that a hotel with infinite rooms has the same size as two hotels of infinite rooms combined. Even better, a hotel with infinite rooms has the same size as infinitely many hotels with infinite rooms combined!

What is Beyond Infinity?

We've developed a way to compare two infinite hotels. Now we are ready to answer our original question:

Is there anything that The Infinite Hotel cannot fit?

The Continuous Hotel

Imagine a hotel that has a room for every real number.

Real number is the fancy mathematical term referring to all numbers that we’re used to. These include the 'counting' numbers like 1, 2, 3 and so on, but also all decimals like 1.9837 and even weird numbers like π or √2.

Think about it: Between any two real numbers, there are infinitely many other real numbers. Between 0 and 1, you have 0.1, 0.01, 0.001, and infinitely many others. This density of numbers creates a different kind of infinity.

If you want the short story, here it is: The Continuous Hotel is larger than The Infinite Hotel.1

If you want the long story, here is a famous argument in mathematics due to the German mathematician Goerg Cantor from 1874.

Cantor’s Diagonalisation Argument (Advanced)

Remember that two infinite hotels have the same size if there is a valid one-to-one matching between their rooms. In other words: two infinite hotels have different sizes if any matching between their rooms is not valid.

The word "any" is the key here. To show that two infinite hotels have different sizes, it’s not enough to show that this or that particular assignment doesn’t work: we need to show that any assignment will not work.

So let’s pick any assignment from The Infinite Hotel to The Continuous Hotel. For every room in The Infinite Hotel—which are numbered 1 to infinity—we have a matching in The Continuous Hotel. It will look something like this:

• Room 1 → Room 1.393849

• Room 2 → Room 78

• Room 3 → π

• Room 4 → 0.66666…

• …

Let’s show that, whatever this matching is, it will miss some room in The Continuous Hotel. I will use the above example to show you the idea, but note that this argument works for any matching:

• Room 1 gets matched to Room 1.393849. Let’s remember its first decimal digit: 3.

• Room 2 gets matched to Room 78. Let’s remember its second decimal digit: 0 (since 78 = 78.000…).

• Room 3 gets matched to Room π = 3,1416…. Let’s remember its third decimal digit: 1.

• Room 4 gets matched to Room 0.66666…. Let’s remember its fourth decimal digit: 6

• …

Now get all these digits and add 1 to them (if the digit was 9, we change it to 0). For example, in the example above we have the digits 3, 0, 1, 6 … and we change them to 4, 1, 2, 7…

Consider the number 0.4127… constructed this way.

This number has not been assigned to any room!

Why?

• It is not the number assigned to Room 1, since its first decimal digit (4) differs to the first decimal digit of the room assigned to Room 1 (3).

• It is not the number assigned to Room 2, since its second decimal digit (1) differs to that of the room assigned to Room 2 (0).

• And so on for every room.

Let’s recap what we’ve done:

1. We considered any arbitrary pairing from the rooms of The Infinite Hotel to the rooms of The Continuous Hotel.

2. We constructed a real number that cannot possibly be among those that the pairing covers.

3. Therefore, given any pairing from The Infinite Hotel to The Continuous Hotel, it misses some room!

Conclusion: The Infinite Hierarchy

What we've discovered is truly remarkable: not all infinities are created equal. The infinity of the real numbers (the "continuum") is demonstrably larger than the infinity of the counting numbers.

But the story doesn't end there. Mathematician Georg Cantor proved that there is an endless hierarchy of infinities, each larger than the last. He used a technique called the "power set operation" to show that for any infinity, you can always construct a larger one.

This leads to a dizzying sequence of increasingly vast infinities, labeled with the Hebrew letter ℵ (aleph):

• ℵ₀ (aleph-null): The size of the counting numbers (0, 1, 2, …)

• ℵ₁ (aleph-one): The next larger infinity

• And so on to infinitely many sizes of infinity

Even more mind-bending, mathematicians are still debating some fundamental questions about these infinities. The famous "Continuum Hypothesis" asks whether there is any infinity between the size of the counting numbers and the size of the real numbers—and amazingly, this question has been proven to be undecidable using standard mathematical axioms!

Infinity continues to challenge our intuition and stretch the boundaries of human understanding. As we've seen, when dealing with the infinite, the ordinary rules break down and we enter a realm where our everyday logic fails us. It's in this breakdown of intuition that we find some of mathematics' most profound and beautiful insights.

What other concepts might lie beyond our current understanding, waiting to revolutionize our thinking once again?

In Case You Missed it

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• The Spanish Christmas Lottery: Dreams, Regrets, and the Reality of Your Odds

• The Limits of Democracy: What Arrow’s Impossibility Theorem Reveals about Fair Elections

• How You Can Make Better Choices (Spoiler: Self-Interest Fails)

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Some weeks ago we explored infinity through the The Impossible Race. The key takeaway was: it is possible to add an infinite number of things yet get a finite result.

I ended that post with a question, which I also asked on a separate poll on Instagram:

The results revealed two interesting insights:

1. People understand that 0.999… is at most 1 (which makes intuitive sense: 0.999... starts with 0, while 1 starts with 1)

2. Many people are uncertain about what 0.999... precisely means: is it equal to 1 or slightly smaller?

Let's Make Sense of 0.999...

The number 0.999… can be written as an infinite sum:

Why is this useful?

Instead of dealing with 0.999..., a concept we might find confusing, we now have a sum of infinite terms where we understand each individual piece.

We still have to figure out what an infinite sum really means, but at least we know the components that make it up.

Two Key Insights

1. If I stop the infinite sum at any finite point, the result will be less than 1.
   Whether it's 0.9, 0.99, 0.999, or 0.9 followed by a million 9s—as long as the number of 9s is finite, the result remains less than 1.

2. I can get as close to 1 as I want by adding sufficiently many terms.
   Want to be within 0.00000001 of 1? Just calculate up to 0.99999999. Need even more precision, like 0.000000000000001? Then calculate up to 0.999999999999999.

Let's visualize these two statements:1

1. Stopping the sum at any given point (the green points below) gives something smaller than 1:

2. Given any maximum tolerance (the orange line below), we can get closer to 1 than that by adding sufficiently many terms:

Why 0.9999… = 1

Now we can make a compelling argument that 0.999... must be equal to 1, not smaller.

Take any number smaller than 1, say 0.999999876. We've established that by considering sufficiently many terms in our infinite sum, we can get closer to 1 than this number (for example, 0.9999999 is closer to 1 than 0.999999876). Our infinite sum 0.999... is larger than 0.9999999 (it's 0.9999999 plus something else), so it must also be larger than 0.999999876.

Therefore:

Given any number smaller than 1, the number 0.999… is larger than that number.

If 0.999... is larger than every number smaller than 1, and it's not larger than 1, then it can only be equal to 1!

If this is still troubling to you, think again about The Impossible Race. Just as Achilles eventually catches the tortoise despite being behind the tortoise at every step in the paradox, 0.999... equals 1 despite 0.9999999 being smaller than 1 for any finite number of 9’s.

Real-Life Applications of Infinite Sums

This concept isn't just mathematical curiosity—it has several practical applications:

1. Computer Graphics: When rendering realistic 3D images, computers use infinite series approximations to calculate lighting and reflections. By adding more terms, the image becomes more realistic, but at some point, adding more terms makes no visible difference—similar to how 0.99999999, with sufficiently many 9’s, is practically indistinguishable from 1.

2. Financial Calculations: When calculating compound interest over many periods, the formula involves a sum that approaches a limit. Understanding convergent series helps financial analysts make accurate long-term predictions.

3. Signal Processing: When your smartphone processes audio or your streaming service compresses video, they use mathematical techniques based on infinite series to approximate complex waveforms efficiently.

Why This Matters

Understanding that 0.999... = 1 helps us grasp a fundamental concept in mathematics: limits.

A limit describes the value a function or sequence approaches as it extends toward infinity. This concept underlies calculus, which powers everything from rocket science to economic modelling.

It also teaches us something profound about infinity itself. Our intuition often fails when dealing with the infinite—what seems "obvious" (that 0.999... should be less than 1) turns out to be incorrect when we apply rigorous mathematical thinking.

A Final Thought

Mathematics often challenges our intuition, pushing us to think beyond what seems immediately obvious. The equality 0.999... = 1 reminds us that rigorous mathematical reasoning sometimes leads to conclusions that initially feel wrong but reveal deeper truths about numbers and infinity.

What other mathematical concepts do you find counterintuitive? Let me know in the comments!

In case you missed it

• The Linda Problem: When Our Brain Tricks Us Into The Impossible

• The Hidden Psychology of Change: How We Value Gains and Losses

• Bitcoin 101: The Secret Power Behind Decentralized Electronic Money

• Bitcoin 201: Why It Compares to Gold & What Happens in 2140

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