Some of the most popular pages on this site don't find one thing in π, they find two. A couple or a pair of besties type in their names and see how many digits lie between them. In a Name Fight, two names step into the ring and whichever turns up first in π wins.
It's all just for fun, but there is real maths underneath, and it explains a few things people notice once they start playing.
From a name to a number
First, a name has to become digits. We use the letters on a phone keypad: A, B and C are on the 2 key, D, E and F on the 3, and so on up to W, X, Y and Z on the 9. So every letter becomes one digit, and a name with five letters becomes a five-digit number. Then we look up where that number first appears in π, exactly like any other search.
That one fact, one letter is one digit, explains almost everything that follows.
Why short names win
Imagine π's digits as rolls of a ten-sided dice (they behave exactly like that in every test anyone has tried). A particular three-digit number has a one in a thousand chance of starting at any given spot. A five-digit number has a one in a hundred thousand chance. So the three-digit number usually turns up about a hundred times sooner.
That's why short names win name fights so often. Here are the typical depths at which a number of each length first appears:
- 4 digits (a name like ANNA): around 7,000 digits in.
- 6 digits (a name like OLIVER): around 700,000.
- 8 digits (a name like ISABELLA): around 70 million.
How often does the shorter name win? There's a neat formula. If one name has k letters and the other has m, the shorter name wins roughly 10^m out of every 10^k + 10^m fights. For a name just one letter shorter, that is about ten fights out of eleven. Two letters shorter: about a hundred out of a hundred and one. Same length? A fair coin toss.
So in a Name Fight between SAM and ALEXANDER, SAM is almost certainly going to win. That isn't a judgement on anyone. It's just that SAM is a three-digit number, and ALEXANDER is a nine-digit one.
How far apart is a couple?
Now for the couples page, which shows the distance between two names: how many digits you would have to walk to get from one to the other.
You might expect the gap to be small, since the names are "near the start". But here's a surprising fact: for two names of the same length, the typical distance between them is about the same as how deep each one typically sits. Two four-letter names usually sit around 7,000 digits apart. Two six-letter names, around 700,000.
Why? The dice model has a strange property called being memoryless. Wherever the first name turns up, the dice don't remember it. From that spot, the wait for the second name is just as long as if you had started from the beginning. So the gap behaves like a fresh search, and has the same typical size.
When the names have different lengths, the longer name decides nearly everything. If one name is four letters (found within a few thousand digits) and the other is eight (found tens of millions of digits in), the distance is basically just how deep the long name is.
So what does our distance mean?
Absolutely nothing about your relationship, we promise. A huge distance doesn't mean you're drifting apart, and a tiny one doesn't mean you're soulmates. It mostly depends on how many letters your names have. Two people called ED and JO will nearly always look closer in π than ALEXANDRA and FRANCESCA.
That's also why the couples and besties pages only ever show the distance, not where either name is. A position in π can give a name or a number away, so we never show it, never store it, and never put it in a link.
Fun experiments to try
- Fight your first name against your surname. Which is shorter? Did it win?
- Try a friend's name with and without its last letter. How much closer does it get?
- Find a couple whose names have the same number of letters. Is their distance close to the typical one above?
- For a longer adventure, write a secret message in π with a Love Hunt.
Sources
- Wikipedia: Exponential distribution, including the "memoryless" property.
- Wikipedia: Geometric distribution, the maths of waiting for a dice roll.
- Wikipedia: Telephone keypad
- Wikipedia: Normal number