Cats And Dogs

Hmm... been some time since I blogged, something of a commentary on the complete lack of interesting puzzles I've come across in the past while. A few months ago I was lying in bed thinking about tic-tac-toe and my mind wandered into a bit of a neat game.

When I was young, it was common for a game of tic-tac-toe (which I will start calling TTT, to avoid typing that out everytime) that ended in a tie to be referred to as "cats", and I remember one friend of mine who would occasionally call a tie game "cats" only sometimes and "dogs" at other times. It took me some weeks of playing games with her before I managed to decipher the rule that she used in calling some games "cats" and some "dogs". Basically it came down to if there was a letter "C" in the final position. Consider a game of TTT with X going first (when I was young, O always went first, but I have discovered in my adult life that I am apparently the only person on the planet who thinks that O goes first, so I will submit and use the majority convention here), if the game ends in a tie there will be 5 X's on the board and 4 O's, and those 5 X's can form one of only three possible shapes (they aren't hard to find, and I don't feel like figuring out a good way to draw them). One of those shapes looks like a "C" when oriented correctly, and the draw would be called "cats" if that shape appeared, and the draw was "dogs" otherwise.

Lying in bed, I considered using this "cats" vs "dogs" distinction as a tiebreaker on the game of TTT. Since it is trivial for X to achieve "dogs" by making their first move in the center, it is natural to break ties as "cats" is a win for X and "dogs" is a win for O.

Looking at the possible tie configurations, we can see that "cats" occurs if and only if X controls three of the four side squares, which basically tells us that the game is about controling the side squares. So we can reword the game as "play TTT as normal, X going first, if the game ends in a tie, then whoever controls more of the side squares wins, if that is a tie then O wins." Actually this game is somewhat interesting, sides are the worst places in normal TTT, but this puts an extra emphasis on controlling the sides, so there is some sort of balance. You want to take the sides as soon as possible, but make sure not to let the opponent get an old-fashioned three in a row while you are doing that.

Anyway, I'm going to leave you with that. If you are like me, you will take the time to solve out this game, its only slightly harder than TTT, and its sort of fun. As adults we never get to solve out TTT because we solved it so young, its nice to be able to start over again.

James suggested calling this game "Cats and Dogs," which seems like a smart enough thing to do, so I guess that will be that.