How many people need to be in a room for two of them to share the same birthday? You are probably thinking something in the hundreds. But for the chance to be greater than 50%, all you need is 23 people!

Not impressed by that? How about a paradox from Plato? A man wants to cross a bridge and Plato tells him he will only allow it if his first statement is true. Otherwise, he will be thrown into the lake. The man tells Plato: “you will throw me into the lake.”

So, what should Plato do?

We’ve compiled a list of problems to flex your mental muscles. From the friendship paradox to the infamous “catch-22”,

**Here are 25 of the most mind-blowing paradoxes of all time!**

## The Drinker Paradox

The Drinker Paradox goes like this: if someone in the pub is drinking, then everyone in the pub is drinking. The problem comes from the wording, which makes it seem that one person in the pub is causing everyone else to drink. By instinct, we know that this is not the case. But when made into a formal logical statement, it means focusing on a person at the pub drinking will prove the statement to be true.

That one person could be anyone, at any different time, meaning there is a lot of flexibility. Either way, whoever is thinking about these kinds of problems at the pub is probably drinking by themselves!

## Simpson's Paradox

Sorry to get your hopes up, but this has nothing to do with the TV show. Instead, the Simpson’s Paradox refers to a strange behavior in some statistics when data demonstrates a pattern, but it is negated when groups are added together. For a simple illustration, let’s say that two people are doing archery, shooting at targets and they are both given two equally long sessions.

If Person A shoots on target more times than Person B. It doesn’t necessarily follow that Person B has had a higher shooting accuracy over the two sessions. How could that be so? Say Person B shot two out of two targets in session one and five out of eight targets in session two, for a total of 70% over the two days. Person A, on the other hand, shot seven out of eight targets in the first session and one out of two in the second session for a total of 80%, higher than Person A. This might sound like abstract thinking, but it’s pretty important in statistics, especially when medical studies where different sections of the population might have different results to the overall population. Moral of the story: stats can be deceiving!

## The Ship of Theseus Paradox

This paradox dates back to the deep thinkers of Ancient Greece. We are asked to imagine a ship, called Theseus, that the government is desperate to preserve. To do that, they replace any pieces that are damaged or falling apart. Slowly, over time, every piece of the ship is swapped for a new one.

The question is: at the end of this process, is it still the same ship? The paradox relates to identity and applies to you too. Every cell in your body is regenerated, just like the ship, so what’s saying that you are the same as a few years ago?

## The Paradox of Social Interaction

Some less than others, but we all need to socialize. And when we have good interactions, they are based on both sides having a common understanding. But where does this process begin? If you try to predict what someone else is thinking or feeling, act accordingly.

But when the other person is doing the same, it creates an endless loop. Even just describing this is enough to give me social anxiety.

## Two Envelopes Paradox

Say that I hand you two envelopes and tell you that they both have money in them. The only difference is that one has $10 and one has $20. You make your choice and then I give you another option — you have one chance to switch the envelopes. Intuition seems to tell us that there is no advantage to swapping. You had a one in two chance going into it and why would anything have changed? Strangely, mathematics says that you should change.

The calculation finds that the value of switching is optimal because you stand to possibly gain double what you have or to lose half of what you have. On its face, we know that this equation is telling us something wrong and it takes some tweaking to get the calculation to arrive at the correct answer, which is that there is no advantage to swap. Sometimes, it’s better to trust your instincts!

## The Birthday Paradox

How many people would have to be standing in a room for someone to share the same birthday as you? Think about it. Immediately, you’ll probably start to calculate the 365 days in the year and work out a formula. But if you haven’t heard this problem before, get ready to have your mind blown. It turns out that for the chance to be over 50%, you only need twenty-three people in a room.

That means that on a normal NFL team roster of 50 players, there should be a 97% chance of two sharing the same birthday. How does it work? To put it simply, it’s because the calculation uses every single possible pair, so for 23 people, it’s 23 multiplied by 22, which is already 253 pairs. Ever wondered if multitasking is actually effective? That’s coming up at number 15.

## The Paradox of Aging

Okay, let’s get out of mathematics and into the real world. This next one is called the “paradox of aging” and depending on how old you are, you might have already experienced it. It refers to the [observable](https://pubmed.ncbi.nlm.nih.gov/33626994/#:~:text=The “paradox of aging” refers,their happiness does not necessarily.) fact that the more a person’s physical and cognitive health declines through age, the happier they tend to be.

This seems pretty counterintuitive when you think about it since our youth is when our body is at peak performance. But it’s only as we age that our well-being increases, which is generally associated with an improved ability to regulate emotions. See, there is at least one positive to aging!

## The Paradox of FOMO

You might not have experienced the age paradox, but you will know what FOMO feels like. That stands for the “fear of missing out” and it’s difficult to escape in the social media age. Whether it’s an event, promotion, achievement, or relationship, scrolling through your feed each day can easily lead to this feeling.

The paradox is that the more we think about the world out there and compare ourselves, the less we value the great things already in our lives and the present moment. Most of us have a lot to be thankful for and we miss out on it by chasing the things we feel we are “missing out” on.

## The Paradox of Happiness

This next one is similar to FOMO. The paradox of happiness is that the more you try to be happy, the less successful you are. Study after study confirms that aiming towards happiness is the worst way to achieve it. That’s because it isn’t realistic to believe that life gets progressively better.

Instead, there are highs and lows, plus, nothing will ever truly be perfect. The way out of this is to stop thinking about happiness so much. But that’s easier said than done.

## The Monty Hall Paradox

Congratulations, you’ve been invited to a game show! The name of it is Monty Hall and you have the chance of winning a brand new car. All you have to do is guess the right door out of three. The bad news is that behind two of them is a goat, in other words, a loss. Since you can’t see behind them, it’s a simple 1 in 3 shot. But there’s a twist. After choosing your door, the host opens one of the remaining doors and reveals a goat.

So, now only two doors are remaining and you’ve already picked one, the question is: should you switch? Most people choose to stick with their original choice. But you’d be wrong to do so. When you made that choice, you had a one-in-three chance. After one of the doors opened to reveal the goat, one of the losing options was removed. Switching increases your chances to 2 in 3 — the key is that you have been given an extra piece of information, so you should use it!

## The Paradox of Multitasking

How many things can you do at the same time? Many of us have great confidence in our ability to do two — or three — tasks simultaneously. Facebook on one tab, work on another, and maybe a podcast in the background. Especially when it comes to work, multitasking is rarely productive.

Trying to spread your attention too thin lowers productivity and can make you more stressed. While it may feel like you are getting more done, the reality is probably the opposite.

## The Gabriel's Horn Paradox

The Gabriel’s Horn paradox involves an object that is infinite in surface area but finite in volume. In the commonly used example, the object is shaped like a horn, and we are asked if it is possible to ever paint the inside of the shape.

The problem comes from the fact that we are trying to compare a two-dimensional number, the surface area, with a three-dimensional number, the volume of it. In reality, it means there is no paradox with Gabriel’s Horn. But that hasn’t stopped it from becoming an important part of mathematics courses. If you like that one, wait until we get to Shrodinger’s Cat at number 10, a problem that lies at the heart of quantum mechanics.

## The Unexpected Hanging Paradox

Let’s move to a paradox that’s a little less abstract. A prisoner has been convicted of a crime and sentenced to hanging. Now, the judge tells the offender that the day of the hanging will be a surprise to him, all he knows is that it will be on a weekday in the following week and he will be notified by a knock on his cell door. Now comes the puzzling logic. As the prisoner begins to work out the judge can’t wait until Friday, otherwise, it will be obvious on Thursday afternoon that the hanging will be the following day, thus ruining the surprise. But the same goes for Thursday, since Friday is out of the question, and on Wednesday afternoon he can make the same assumption.

In fact, this is the case for each day, working back to Monday, leading to the conclusion that there is no way for the prisoner to be surprised at all. The paradox comes from how deductive thinking can arrive at an incorrect conclusion. If he hears a knock on the door on Tuesday, the prisoner will no doubt be surprised.

## The Friendship Paradox

How many friends do you think you have? And out of those friends, do you think you have a bigger or smaller network? Most people think they are the connector, but according to statistics, it’s more likely that you have a smaller friend group and people you know have a bigger one. That’s because of the way that network structures work.

You have a much lower chance of being friends with someone with very few friends than someone with a lot of friends. I’m gonna go and check my Facebook account right now!

## The Paradox of Control

Just like the paradox of happiness, this one is about achieving what you truly want. In some situations, the more we try to control things, the worse the outcome is. Psychologists call this “misapplied control” and it’s a natural tendency.

The instinct to control things often improves our lives, but focusing on every aspect is counter-productive, where we waste our energy worrying and becoming anxious. Letting go of control and accepting the risk of failing can actually lead to better results — and a better life.

## Paradox of Schrödinger's Cat

This next one is a good illustration of quantum physics. Don’t worry though, there isn’t any math or calculations involved. It’s called “Shrödinger’s Cat”, named after the physicist who thought it up. The basic premise goes like this: you put a cat into a box alongside some radioactive substance. If the atoms from this substance decay, then it will cause acid to be released and kill the cat.

But the paradox of the scenario is that you cannot tell if the cat is dead or alive until the box is opened. And because of the probabilistic way that the radioactive material works — in what’s called a “psi-function”, the cat can be described as being dead and alive at the same time. The thought experiment shows the strangeness of quantum mechanics, specifically “superposition”, which is when something can be in two states at the same time. I have a question though: did anyone check on Schödinger’s pets though?

## Paradox of Information Overload

You know that feeling of standing in front of a supermarket aisle, paralyzed by a mountain of choices? That’s not just you. Research tells us that we can only hold between five to nine items of information at a time. This forms the foundation of the paradox of information overload.

We usually think that more information is better, but psychology proves that we are only able to use a very limited amount of it. It’s much better to focus on a few key pieces rather than try and take in everything all at once.

## The Paradox of Fiction

When was the last time you read a good book or watched an engaging movie? What made it good was probably how it engaged you emotionally. But there’s a paradox here, composed of three parts. Firstly, most of us experience emotions drawn from works of fiction. Secondly, we need to in some way believe in the story that is being presented to us. But finally, nobody really thinks that the characters they are watching on screen are real.

So, how can all three of these things be true at once? One answer to this is that humans are able to build a kind of emotional distance between ourselves and the world of fiction so that we experience a less intense version of real emotions.

## The Banach-Tarski Paradox

Take a hard ball and pick apart its outer shell into a handful of pieces. According to the Banach-Tarski Paradox, you can then reassemble these pieces into two balls that are the same size as the first ball.

How is this possible? The answer lies in the mathematics used, which allows for a finite number of pieces to be re-assembled into a shape that is even bigger than originally, but that doesn’t maintain the same volume.

## Meno Paradox

Meno’s paradox is the paradox that if you don’t know what you are looking for, it is impossible to find it. This problem dates back to the Ancient Greeks and Socrates. It’s an important problem to address because if it’s true, then increasing our knowledge into new areas and learning seems to be blocked. All science then becomes pointless.

Socrates resolves this by arguing that there is a kind of knowledge that we are unaware of, deep inside of us, that has to be accessed for learning to occur.

## The Raven Paradox

Here’s another paradox that poses a problem with science. Say that you begin to notice that all of the ravens you see are black. From these observations, you can begin to build a hypothesis that “all ravens are black.” This is a different way of saying that if something is a raven, then it’s black. Then, we should be able to switch the two parts of the statement like it was an equation.

In philosophy and logic, this is not uncommon. But then we run into a problem because the statement now claims that “if something is not black, then it is not a raven.” There are plenty of things that are not black that are not ravens either. This might sound like wordplay, but academics have been proposing solutions for hundreds of years — and there’s still no agreement!

## Zeno's Paradox

You might not know it, but anytime you put one foot in front of the other and move, you are proving one of Zeno’s paradoxes wrong. These are thousand-year-old problems that seem to go against all our common sense and the world around us. The paradox of motion goes like this: you are in a race against a tortoise. This should be pretty simple until Zeno breaks it down for us.

To travel 100 feet, we have to travel 50 feet and to do that, we have to travel 25 feet. This chain of reasoning goes on forever, literally. It turns out there are an infinite series of steps and the remarkable conclusion is that we can’t move at all. Obviously, we know this to be false, but that’s what makes it a paradox!

## The Sleeping Beauty Paradox

Something named the “Sleeping Beauty Paradox” may not sound serious, but it has stumped mathematicians for decades. It starts like this: you are in an experiment and put to sleep after being told that you will be woken up either once on the following day or put straight back to sleep for the next couple of days. This number depends on whether the toss of a coin lands on heads or tails — heads for one and tails for two. After three days, the experiment ends. Now, I know this is confusing, but bear with me. When you are put to sleep, you have no memory of how many times you have been put to sleep.

The question comes when you are awoken: what is the probability that the coin was flipped to heads? Most people would think the chance is 1 out of 2. But there is a case to be made for guessing that it is 1 out of 3. Why? Well, you have no idea what day it is or how many times you’ve been put to sleep. There are three possible outcomes: either a head was thrown and you are woken up the following day. Or a tail was thrown and you are experiencing the first day or the second day. Whichever one feels right, don’t worry, you are not alone, plenty of very smart people are positioned in both camps.

## Catch-22 Paradox

Ever heard the term “catch-22”? You probably know the broad meaning of it, but the problem is more interesting than you might know. The origin of the term is from a 1960s novel where someone is trying to get out of military service. The conditions state that someone has to be “mentally unfit” to be relieved from duty.

But here lies the paradox: if you ask to leave, then it demonstrates an awareness of the dangers of the situation, which is rational, and also means that there is no way to escape service.

## Buridan's Bridge

Let’s take it back to Ancient Greece one last time. In this paradox, Plato is guarding a bridge when his greatest student Socrates wants to cross. Plato gives him one condition to pass through safely. He tells Socrates that if the first statement he makes is true, then he will be allowed to cross unimpeded. But if he lies or says something untrue, then Socrates will be pushed down below into the water.

Known to be cunning, Socrates answered by saying: “You will push me into the water.” Now, you probably see the paradox. If Plato pushes him in, then it will turn out that Socrates’ first statement was true. Or, if Socrates is allowed to pass, then once he has passed the bridge, his first statement will turn out to be untrue. There are several proposed solutions to the paradox, but by far the funniest is letting Socrates pass before pushing him in any way. That’s what he gets for trying to be too clever!