A few days ago I played the following version of a classic two person take-away game with Katie: There are n chips (we started with 5 :-). On your turn you can take one or two of them. Whoever takes the last chip wins. If you are not familiar with this type of game, then you might want to figure out for yourself what the winning and losing positions are before reading on.
I start by asking Katie whether she wants to go first or second. First, of course; what a silly question. She proceeds to take one chip and I follow by doing the same. Katie looks at the three remaining chips and then back at me. “Heeeyyy,” she says, as she realizes that she’s in a losing position. Naturally, we decide to play again. This time Katie wants to go second. I take two chips, leaving her again with three. And again, “Heeeyyy.”
On take three, Katie first isn’t sure whether she wants to go first or second, but after thinking about it for a bit she decides that she’ll go first. She then takes two chips and gives me a victorious look.
Me: See, you won.
Katie: But you won twice and I only won once.
Me: In this game it doesn’t matter how many times you win. The important thing is that you understand how to win.
Since Katie is willing to continue, I add another chip to the pile and ask the usual question. Katie chooses to go first, and of course loses. On the second try, however, she decides to go second and wins.
Finally, I make the game a bit harder by adding four more chips, for a total of ten. Remembering the previous time, Katie decides to go second and gives me another “heeeyyy” when she loses. When next time she picks to go first and figures out the steps to win, I am pleasantly surprised. Although perhaps the first few decisions were just based on luck. I’m sure that she doesn’t understand yet what’s going on, but I think that she started getting a feel for the game. At the end I was also pleased to hear her say that she enjoyed the game and wanted to play it again some time.
Musings of a Mathematical Mom 