Superpermutation

Okay, so you know how sometimes you have a bunch of letters and you can make different words out of them, like cat, hat, bat? Well, sometimes people like to play a game where they try to make a really long word that has all the different combinations of those letters.

A superpermutation is kind of like that, but with numbers instead of letters. It's a really long sequence of numbers where every possible combination of a smaller set of numbers is in there somewhere.

So let's say we only have two numbers, 1 and 2. The superpermutation for those numbers would be 121, because it has both 1 and 2 in it, plus all the different ways you can combine them: 12 and 21.

But what if we had three numbers? Now we need to make sure we have every combination of three numbers in our sequence. So we start with our first number, 1. Then we add the second number, 2. But now we need to make sure we have 1 and 2 together in both orders, so we add 12. Then we add the third number, 3. We need to make sure we have every possible combination of 1, 2, and 3, so we add 13 and 23. Then we need to make sure we have all the combinations of 1, 2, and 3 with 12, so we add 312 and 231. And we keep going like this, adding numbers to make sure we have every possible combination.

As you can imagine, it gets really complicated when we start adding more and more numbers. But people have found ways to figure out what the shortest superpermutation is for different sets of numbers. It's a tricky puzzle that keeps mathematicians busy!