Maths is weird. The most absurd facts about maths require advanced knowledge of pure mathematics but there are plenty of oddities that all of us can grasp. You’ve maybe typed 58008 into a calculator, or wondered what happens when you multiply 111111 by 111111, or noticed that pi to 3 significant figures (3.14) is “pie” backwards. We’re going beyond little calculator tricks today but avoiding the most advanced mathematics. Here are our 7 favourite maths facts that break our brains.
1. Probabilities and shuffling cards
If you shuffle a deck of 52 cards, you will likely end up with an order of cards that nobody else has ever had before in the history of people shuffling cards. Seriously. There are only 52 cards and people have used cards for hundreds of years, so what gives?
This whole idea assumes everyone shuffles the cards properly so they’re in a truly random order. If you shuffle them thoroughly, the top card on the deck has a 1 in 52 chance of being the Queen of Clubs or any other card. As for the next card down, there’s a 1 in 51 chance it could any given remaining card. The same goes for the 3rd card, which has a 1 in 50 chance of being any given remaining card.
There’s nothing astounding about any of that but things escalate when you think about those 3 cards together. The odds that you would get three specific cards in a specific order (e.g. Queen of Clubs, 2 of Hearts, and 7 of Diamonds) if you did it all again are quite low. There are 132,600 possible combinations of cards you could get as the first 3. When you add a 4th suddenly there are 6,497,400 possible combinations. When you do this for a full deck of 52 cards, there are 8.0658X10⁶⁷ possible combinations. That number is 68 digits long.
How many times have decks of cards been shuffled in history? If we assume that every single person on Earth has been shuffling decks of cards every second for the last few hundred years (which is a massive overestimation), the number of times doesn’t come anywhere close to the number of possible combinations. So when you shuffle a deck of cards thoroughly, it’s extremely likely that nobody else has ever shuffled a deck and got the same order of cards from their shuffle.
Bizarrely, the probability that any two shuffles in history were the same is higher than the probability that your shuffle matches one in history. Enter the birthday paradox…
2. Birthday Paradox
The birthday paradox has some complex maths behind it and explains the weirdness of the above point. There’s a certain probability that your cards will match someone elses but it’s lower than the probability that any two in history match each other. This seems to be counter-intuitive and leaves many people scratching their heads.
The paradox gets its name from an even better example: imagine you walk into a room with 22 other people already inside. The probability that anyone in there has the same birthday as you is really low. Indeed, it’s likely nobody there shares your birthday. However, the chances that any two people in the room share the same birthday is 50%.
This seems like a paradox… aren’t you still comparing two people when you look for someone who shares your birthday? It isn’t a paradox though, it’s just weird and it’s why it’s more likely that two people in history have shuffled the same combination of cards than you shuffling the same as someone else in history.
The reason this happens is that you’re drastically changing what you’re actually looking for. If you just need two people with the same birthday, you’ve got a 50% chance of finding a pair in a room of 23 people. Up the number of people to 75 and it becomes a 99% chance that a pair share a birthday. But when you look for someone sharing your own birthday, you’re actually already thinking of a specific date, so the odds are 1/365. It’s much harder to find two people who share a specific day we’ve already decided, as opposed to simply sharing any day.
This is why the above shuffling example has a supposed paradox. It’s hard enough for two people to identically shuffle any combination of cards. To also have that shuffle be the exact same as yours should be even more unlikely as you’ve added restrictions.
3. 0.9999 etc = 1
In maths, a repeating decimal like 0.99999 (and so on) literally equals 1. They’re the same number. This is actually true for any definite value so 1.99999 (and so on) equals 2. Don’t believe us? Try it out:
x = 0.9999 etc
That means 10x = 9.9999 etc
Now subtract x from 10x, which is 9.9999 – 0.9999 = 9
So 9x = 9
This means x = 1, so 0.9999 = 1
4. Monty Hall paradox
This is another example of probabilities messing with our heads. Indeed, our brains are so biased towards thinking a certain way that some expert mathematicians have refused to accept the Monty Hall paradox until seeing computer simulations proving it to be true.
Imagine being on a game show where you’re shown 3 doors and have to pick the correct one to win a prize. Let’s say there’s a cash prize behind one door but nothing behind the other two. If asked what your odds are of picking the correct door, most people would probably say 1/3 and that’s correct. Nothing weird about that.
Let’s say you choose door number 1 but before you get to see what’s behind the host suddenly opens door number 2 to show you that there’s nothing there. They then give you the chance to change your mind. Do you want to stick with door number 1 or change to door number 3? Most people think it’s best to stay because your odds of being right have seemingly shifted from 1/3 to 1/2. Counter-intuitively, this is actually incorrect and you should definitely switch.
The mathematical weirdness of this problem means that you will improve your odds by changing your mind in this situation. It’s absolutely correct that the odds of choosing the right door in the first place were 1/3. But now that you know one of the incorrect doors, the odds haven’t changed to 1/2. Instead the odds for your door remain at 1/3. What does change is that the other door (number 3 in our example) has seen its odds shift from 1/3 to 2/3. It’s difficult to understand why, but the odds of number 3 are now higher than number 1. It may help to think of doors 2 and 3 together having a 2/3 chance of being right at the start. Now door 3 is the sole representative of those two doors and still has a 2/3 chance, better than the 1/3 chance of your original door.
5. Kaprekar’s constant
There are loads of weird numbers that have some odd properties. One of our favourites is 6174, also known as Kaprekar’s constant. Grab a bit of paper and try this:
- Think of any 4-digit number that has at least two different digits (e.g. 1112 is allowed, 1111 is not).
- Now arrange the digits in descending order and then again in ascending order giving you two different numbers.
- Subtract the smaller number from the larger one.
If you do this you will soon get Kaprekar’s constant: 6174. It doesn’t matter what 4-digit number you start with, you’ll always get to 6174 and then get stuck there because 6174 itself gives you a result of 6174. Weird, huh? 495 is the 3-digit equivalent.
6. Benford’s law
If you take real-life data such as heights of buildings, lengths of rivers, populations in countries etc you find that there’s a bias towards the first digit being a low number. It’s more likely to be 1 than 9, for example. The law’s Wikipedia page gives the example of the top 60 tallest buildings in the world. It doesn’t matter if you use the height measurements in metres or in feet, the first digit is more likely to be a low number.
The law works best for data that’s distributed across orders of magnitude. So it doesn’t work well for, say, the heights of people. But across orders of magnitude it holds true a surprising amount of the time. There’s a mathematical explanation but intuitively it seems weird that 2 or 3 should be more common than 7 or 8 as a first digit when measuring things like the lengths of rivers.
The law works so well that it has been used to detect fake numbers in tax documents. When people make up fake numbers, they try to make them realistic and avoid bias towards certain numbers. But as Brenford’s law points out, the lower numbers are often more common as first digits. Studies have been able to find fudged data by looking for tax returns that don’t fit with the law!
7. Mandelbrot set
Maths is beautiful but sometimes it isn’t obvious to those of us without advanced knowledge. The Mandelbrot set allows the beauty of maths to be obvious to everyone. It’s really easy to make a formula that makes numbers escape to infinity. The Mandelbrot set is a collection of numbers that you can plug into a simple formula (z -> z² + c) and they don’t reach infinity.
If you represent the numbers and formula graphically, you get spectacularly weird images that are surprisingly complex. If you zoom in on the Mandelbrot set you see more Mandelbrot sets. Zoom further and those sets contain more sets and this process does indeed go on forever. Within the simplicity of z -> z² + c you can see infinite complexity.
These words don’t do it justice so watch this video that zooms into a Mandelbrot set. What you’re about to see hasn’t been deliberately designed by anyone; the patterns are dictated by that very simple formula.