9-square fold solutions

See the original puzzle

In order to keep the different squares straight, I'm going to label each one with a number.

Now, I could just give you the solutions by giving the correct order of numbers. For example, the "spineless" one is 743652189. But that's kind of boring, and it doesn't really tell you exactly how you're supposed to fold it anyway. Instead, I'll discuss the general principle behind the 9-square fold.

The general principle is that paper can't go through itself unless you tear it. Try poking your finger through paper without tearing it. It doesn't work. Simple, eh?

Let's consider a simpler case: a 4-square fold with just the numbers 1, 2, 4, and 5. Can we create the following stack?

1245

There are exactly 4 folds in every 4-square stack. Those folds are between squares 1 and 2, 2 and 5, 4 and 5, 1 and 4. I will call these folds 1-2, 2-5, 4-5, and 1-4 respectively. If you create a square stack, you will find that folds 1-2 and 4-5 are always on the same side of the square, while 1-4 and 2-5 are on another side of the square. That means that the links 1-2 and 4-5 cannot cross each other, nor can the links 1-4 and 2-5. Paper can't go through itself! In the fold 1245, 1-4 and 2-5 cross each other, therefore it is impossible.

For a 9-square fold, there are 12 folds: 1-2, 2-3, 4-5, 5-6, 7-8, 8-9, 1-4, 2-5, 3-6, 4-7, 5-8, 6-9. Remember, the final stack will be a square with four sides. One side will have the folds 1-2, 4-5, and 7-8. Another will have 2-3, 5-6, 8-9. Another will have 1-4, 2-5, 3-6, and the last will have 4-7, 5-8, 6-9. Allow me to put this into a table:

Within each of these groups, there can be no crossed folds. This is sufficient to distinguish between possible and impossible solutions. Now you can tell, at a glance, whether the sequence 123456789 is possible (it's not). Of course, this still doesn't help you actually fold the paper. That is something that can only be mastered by trying it yourself.