For this week, we’re going to look at two very different activities that show a surprising result when presented together. Puppies and Kittens was first presented by Paul Zeitz for the Math Teachers’ Circle Network, and Queen’s Move is more commonly known as Wythoff’s game.
Puppies & kittens
A local animal shelter has a certain number of puppies and kittens available for adoption that you just happen to be itching to own! In this week’s “paw-some” activity, two players begin with a certain number of animals to choose from. Each turn they adopt more by following one of these rules:
- adopt only puppies (no kittens)
- adopt only kittens (no puppies)
- adopt an equal number of puppies and kittens.
The goal is to be the player who adopts the last remaining pet(s)!
It’s a good idea to have new players start the game off with 3 puppies and 5 kittens. This game is pretty quick to play and should be done in a few moments. Play the game multiple times, and let each player get a turn at going first. The interesting part of the game is less about winning a single game and more about the following questions:
- Is there a strategy for winning the game?
- Does it matter who goes first? Does one player have an advantage?
After players try a few games with 3 puppies and 5 kittens, mix it up! Let the players decide the number of each animal to start the game. To keep things fair, let one player decide the number of each animal, then let the other player decide who goes first.
How It Works
To see what’s happening mathematically, it may be a good idea to work on a smaller problem: 2 puppies and 1 kitten. We can map out all possible outcomes (assuming both players are trying to win):
- If the first player takes the same number of puppies and kittens (one each), that would leave one puppy left for the second player to become the winner. (Second player wins)
- If the first player takes both puppies, then the second player is left to choose the kitten and win the game. (Second player wins)
- If the first player takes the kitten, this leaves two puppies left for the second player to take and win the game. (Second player wins)
No matter what the first player tries, the second player is guaranteed a win!
As exciting a result as that is, it’s a very specific case, right? Well, what if the game started with 2 puppies and 5 kittens? If the first player took away 4 kittens, then the other player is left staring at 2 puppies and 1 kitten. And as we showed above, that’s a guaranteed loss, no matter what they choose! Similarly, any game which is one move away from 2 puppies and 1 kitten (or 1 puppy and 2 kittens) is a guaranteed win for the first player. Working backwards, you can get a whole class of starting setups which guarantee a win for the first player. Finding the pairs of puppies and kittens not in this class can show the games in which Player Two is guaranteed to win.
For our second activity, players will be taking turns moving a Queen on a chess board. Their goal is to move the Queen to the bottom left square, and every move must get them to that square. This means each turn, the player can move the Queen in any of these possible ways:
- any number of spaces downward
- any number of spaces to the left
- any number of spaces on its downward-left diagonal.
The game can start with the Queen on any square, so players may want to try the game with many different starting positions. This is another game to play multiple times to to figure out the best strategies.
How It Works
Even though you can start the game on any square, some squares are more fun to play than others. In fact, if we start the game on any square in which the Queen has a single move to get to the bottom left, then the first player can win the game easily. We can see that as such:
Working outward from A1, we see that B3 and C2 are the first squares on which the first player isn’t guaranteed a win, and instead the second player will win no matter what the first player does. We can demonstrate the steps, but this might already be starting to look familiar. If it’s not, let’s change our axes so it’s more obvious.
Rules to Queen’s Move vs. Puppies Kittens
- moving down = adopting puppies
- moving left = adopting kittens
- down-left diagonal = adopting equal number of both
Queen’s Move is a geometric representation of puppies and kittens. Working backwards with the Queen’s movements makes it a lot easier to see which starting positions guarantee a win for either player. This is a great activity to let students and educators make inferences for themselves. It’s recommended that you introduce these as two separate activities and let players note the similarities for themselves.
However, that’s only the beginning of the mathematics you can explore! Once you identify more and more sets of winning positions to Puppies and Kittens, several well known mathematical sequences appear. For a more in-depth look at these games, check out this article from David Austin of the Alliance of Indigenous Math Circles.