Elli recently got a game as a present from her aunt and uncle. The basic idea is that there are 31 cards in the game, each of which has 6 pictures on it. Each picture appears 6 times in total. The way that the game is played is players look at a pair of cards and try to be the first to figure out which picture appears on both cards. The marvellous thing about it is that each of the 31 cards has precisely one matching picture with each of the others. That isn't an easy thing to do - brute forcing that solution would be an incredible endeavour. I suspect that most people would look at such a game and think that it was very clever to have precisely one match for each pairing but not think much about it, whereas I really wanted to know how that would be done.
It turns out that projective geometry makes finding such a solution easy. My father in law did his PhD in projective geometry so he was immediate fascinated by the game and was delighted that he could use his knowledge of a pretty esoteric subject to discuss how the game would be constructed. I have a math degree but I never took a course on projective geometry so my grasp of the subject is sketchy at best. The basic idea is you construct a plane containing the (x,y) coordinates from 0 through 4 to generate 25 points at the intersections and then add 6 more points at infinity to get to a total of 31 cards. The points are the cards and the lines through them represent the symbols on those cards. Using this construction you can find the single possible solution.
What I immediately wondered was which of two things had occurred: Did a math geek look at this and think that a game could be made of it, or did a game designer try to make the game, realize the brute force method was insane, and then contact a math geek to figure it out? It seems likely that one of the two things happened but I certainly couldn't say which. It is cool for me to contemplate though because while I don't actually know projective geometry I certainly *could* know it and I like to design games. This is a thing I could have built.
I *wouldn't* have built it, but I could. That is, while the construction is very interesting indeed the actual gameplay is a simple pattern recognition game. Wonderful for children, actually pretty good for adults competing against children, and of no real interest to me. If I am going to play against Elli this is ideal because I can totally go for it and she has a real chance to win, unlike nearly all other games, but I can't bring myself to care much for a game where there is no strategy.
This is unfortunate I think because games like this that are both elegant and fill a great niche are really useful to the world. It is something I wish I could build, but which I know I never will.