Inspired by the Card Flipping Proof by Numberphile, Simon created his own version of this proof. He made a solitaire game and proved why it would be impossible to solve with an even number of orange-side-up circles. He drew all the shapes in Microsoft Paint, printed them out and spent something like two hours cutting them out, but it was worth it!
If there’s an odd number of orange circles in the middle, then the end pieces are the same, both orange or both white. In both cases the total number of orange circles will also be odd. If there’s an even number of orange circles in the middle, then the ends have to be different (one orange, one white).
In the case of odd number of orange pieces, the ends have to match. In the case of an even number of orange pieces, you would have pieces that point the same way at both ends. “Now we’ve proven that to make this puzzle possible it has to have an odd number of orange pieces”, Simon says.
Why? Imagine a stick figure that always walks to the right, but always faces in the direction of the arrow (as in it can’t go backwards). It would flip every time it reaches an orange circle. Focusing on everything except the ends, if there are an odd number of orange circles, the puzzle pieces would face the other way. Which means that the end pieces are the same, and therefore the end circles are the same. If there are an even number of orange circles in the middle, the puzzle pieces would face the same way. Which means that the end pieces are different, and therefore the end circles are different.
Simon finds this sort of proof easy, but I felt like my brains are going to boil and dripple through my ears and nostrils. He patently exlained it to me several times and types the above explanation, too.