The Puppies and Kittens Game
For this game, two players alternate turns. We start with a pile of 7 kittens and 10 puppies. A legal move is one of the following three possibilities:
removing any positive number of puppies (but no kittens), or
any positive number of kittens (but no puppies), or
an equal number of both puppies and kittens.
The winner is the last player who makes a legal move. Which player has a winning strategy? For what starting values of k kittens and p puppies will player 1 win? For what values of k kittens and p puppies will player 2 win?
Exploring this game required us to build on to our previous experience of the concept of “oasis” and “desert”. An oasis is a place in the game that we can't lose. For example, if we have just had our turn and we have taken all of the puppies and kittens leaving 0 of both, then we win. So, 0 puppies and 0 kittens is a great spot to leave your opponent.
We then explored different combinations to determine if the situation was an oasis for us thereby leaving our opponent in the desert. What if, there were 2 of one and 1 of the other? The first player could not take all of the animals therefore leaving a final move. So 2 of one and 1 of the other is an oasis point for us to leave. Deciding to organize the data in a graph to look for potential relationships helped us to determine the next couple of oasis points. So in order to win, we need to get to one of the oasis points and keep the game at these optimal points.
But why? Once the values are found and put in a table, a pattern can be discerned. The pattern can be generalized and proved using higher level math involving the Fibonacci Sequence. So if you have kids that can take it that far, excellent. Otherwise, simply find a strategy and play to win. Intermediate and Middle School students can still enjoy finding a strategy to win by simply collecting data and finding a way to win every time.
I tried not to give away any secrets. Grab a colleague, a child, or student and discover the math this game has to offer.