Monday afternoon my fellow Math for America colleagues in New Haven had a going away party for myself and another one of us that is moving back to New York City. (She ended up not being able to make it, unfortunately.) We ate hot dogs, drank beer, sat on the beachfront porch at Michelle’s house in Milford, and, of course, played a game: Ticket to Ride. Mike also taught me a new game he learned at the Museum of Math, but I’ll come back to that.

Mike had introduced us to Ticket to Ride once before, also at Michelle’s house, so all of us but Bridget were already familiar with the rules of the game. Last time we played, I won largely from a big connection from New York to Los Angeles, as well as by having the largest network of connected trains. Normally this would have been much harder to accomplish, because with more experienced players, so I hear, there is much more blocking going on, where players try to anticipate and stop each others’ routes they are building. Not so with us. We all pretty merrily went along just trying to complete our own ambitious plans.

I managed to win again from two large connections: Los Angeles to Miami, and Portland to Nashville. (I might have that second route wrong, but it was comparable to that.) Even when I got worried that I was getting blocked off, when it would have been easy for more experienced and aggressive players to cut me off and make my routes nearly impossible to complete, I was able to pull it off. Mike saw what I was doing, but it would have been too difficult for him to stop me by that point. I also saw that he was going for LA to NY, but I didn’t have the time or resources to stop him with what I had ahead of me.

I managed to get the last play of the game and end it with the largest connected network of trains. I think all four of us each had all of our trains connected—again, there really wasn’t any blocking going on—but since I happened to use up the most trains, I had the largest network. This turned out to be critical: if Bridget had been able to take her next turn (right after my last turn), she would have ended the game instead with the largest connected network, swinging the game 20 points in her favor, and she would have won! You wouldn’t have known it necessarily just by looking at the points at the end, but the game was very close.

Beside blocking being something to try focusing more on, there seems to be a clear equivalence in Ticket to Ride: every turn spent drawing cards = two cards (provided you don’t pick a wild) = two future trains on the board. This doesn’t mean two extra points, because longer connections are worth more, but, particularly if you’re after the largest network bonus, it’s worth keeping in mind that, ignoring the extra cards in your hand at the end of the game, every card you draw will be another train on the board.

There’s all kinds of probability modeling you can do with Ticket to Ride with the drawing process. If I need any of X different colors, and none are face up to pick, but a wild card is, should I take the wild card or draw two cards face down? There are 8 colors in the game, so my gut is that X would have to be about 4 before it makes sense to draw face down.

Lastly: the game Mike taught me, which is very simple. Each of two players takes turns picking one of the numbers 1 through 9. No number can be picked more than once, so if I take 6, you can’t take 6 also, and I can’t take it again. The game ends when either player has a combination of exactly three numbers that sum to 15, and that player wins; or it ends when all 9 numbers have been picked without a winner.

I suggest that you think about how you would play this game first, or even play a few matches with a friend (or with yourself), because if you keep reading, I might spoil the whole game for you.

Mike said he believed he had an optimal strategy for the game. I was interested to try to develop a strategy of my own before hearing his, and quickly decided that picking 5 to start would be safe. I was more correct than I knew, because this can lead to never losing, but it might also be too obvious to your opponent what to pick to keep you from winning as well. Mike believed it was better to pick an even number to start, and mentioned something about the even or odd numbers being in the “corners” when you arrange them in a 3 by 3 square.

I had a sudden idea: “I think we might just be playing tic-tac-toe on a magic square.”

It took some remembering (read: Wikipedia) to remember the algorithm for constructing an odd-ordered magic square, but once I made it, it became clear that the optimal strategy for this game of 15 was identical to an optimal strategy for tic-tac-toe, provided you label the squares on the board as a magic square.

I’ve already spoiled enough, so I’ll leave it there.