Doignon's theorem
Doignon's theorem is like a secret code that helps us solve puzzles with numbers! Imagine you have a bunch of different numbers like 1, 2, 3, 4, 5. Now you want to try and put them together in different ways to make new numbers. For example, you could group 1 and 2 together to make the number 12, or you could group 2 and 3 together to make the number 23.
Doignon's theorem tells us that no matter how many numbers we have, and no matter how we group them together, we will always end up with a certain pattern of numbers. Specifically, we will end up with a pattern that looks like a ladder, with the numbers increasing one at a time. So if we start with 1, 2, 3, 4, 5 and group them together any way we want, we will always end up with numbers like 123, 234, 345, and so on.
This might sound a little bit confusing, so let's try another example. Imagine we have the numbers 3, 1, 4, and 2. We could group them together like this:
- 31
- 42
- 13
- 24
- 321
- 432
- 314
- 241
No matter how we put these numbers together, Doignon's theorem tells us that the pattern will always look like a ladder where the numbers are increasing one at a time. So we can take all of these numbers and arrange them in order like this:
- 12
- 13
- 14
- 23
- 24
- 34
- 123
- 124
- 134
- 234
- 1234
This is the pattern that Doignon's theorem gives us, and it works no matter how many numbers we have or how we group them together. Pretty cool, huh?
Doignon's theorem tells us that no matter how many numbers we have, and no matter how we group them together, we will always end up with a certain pattern of numbers. Specifically, we will end up with a pattern that looks like a ladder, with the numbers increasing one at a time. So if we start with 1, 2, 3, 4, 5 and group them together any way we want, we will always end up with numbers like 123, 234, 345, and so on.
This might sound a little bit confusing, so let's try another example. Imagine we have the numbers 3, 1, 4, and 2. We could group them together like this:
- 31
- 42
- 13
- 24
- 321
- 432
- 314
- 241
No matter how we put these numbers together, Doignon's theorem tells us that the pattern will always look like a ladder where the numbers are increasing one at a time. So we can take all of these numbers and arrange them in order like this:
- 12
- 13
- 14
- 23
- 24
- 34
- 123
- 124
- 134
- 234
- 1234
This is the pattern that Doignon's theorem gives us, and it works no matter how many numbers we have or how we group them together. Pretty cool, huh?