The other day, while collecting my childhood belongings from my mother's house, I found my old peg-solitaire set. The following week I was totally obsessed with

The other day, while collecting my childhood belongings from my mother's house, I found my old peg-solitaire set. The following week I was totally obsessed with it, playing it all the time, sharing it with all my friends, and I ended up discovering some really nifty facts about it. I quickly found that as starting positions go, only certain specific choices for the first open hole would make the game winnable (in the sense that you remove all except one peg). For others there would always be at least two pegs left. I also found that the final relative configuration was to a large degree decided (modulo some relative shift) by your choice of initial configuration. After gaining intuition I decided to apply some rigor, so I went about things mathematically and found an invariant! That is, I found a way to compute, from any board state, a number between 0 and 15, which would be the same after any legal move. This meant that there were (at least) 16 groups of board states that were inaccessible from each other by legal moves. And of these 16 groups, only 9 were congruent with a board state containing just one peg. In particular, the octagon board (the one I had as a kid, apparently referred to as the French variant) with the center peg missing in the starting position is not solvable! Once I felt that I was done with investigating the puzzle for myself, I started searching around online and found that my results were (unsurprisingly) far from novel, and were in fact only scratching the surface of what brighter mathematicians had found over the years. But that only made me more happy with my findings, because it confirmed to me that I was right and also that other people were interested in this thing. I've thought about writing an interactive article presenting the findings I made and intuitions I used. Perhaps I'll post it on here once I'm done with it.