Two Player Game Winning Strategy

I had to go to West Edmonton Mall to get my Christmas shopping done. While I was waiting I found myself roaming around the store and came across this game called ‘GoPop!’, now I had just finished writing that morning my final on discrete mathematics. We analyzed various two-player games. This new game was interesting to play with, but more particularly, I was interested in if there was a winning strategy.

The game works like this, we have a 3 x 3 (normally a much bigger board but for simplicity, I am just looking at the mini version) board and in each of the squares is a little popper that flips orientation when it is pushed. The rules are simple, you can pop any number of squares you want as long as they are in the same line or diagonal. The goal is to force your opponent to pop the last square. Meaning when they hand it back to you every square is popped. An example of the first two moves could be as follows.

I will be denoting circles as unpopped squares and empty as squares that have already been popped. This is an example of first a game opening.

The First two moves of an arbitrary game.

First, I want to briefly cover some terminology describing states of the board. A cold/losing board is not what you want. This means that you are in a position to lose. A hot/winning board is a position where there is a way to put the opponent in a cold position. Everytime a player is in a cold position they must put it into a hot position. If they were able to put it into a cold position from a cold position then it would be impossible to win because there would be no winning boards. The only other option is a hot position to a hot position, this would be a silly move though because it would be putting the other player in a position to win.

The first question that I had while playing with this if there was a winning strategy. I started off by assuming that the first player can always win, if that cannot be shown, in particular we show that there is no winning strategy for the first player, we will look at the second player to try to find a winning strategy. I started playing around and I believe that I have found one. An important note is the many symmetries in this game. It can be reflected vertically, horizontally, or diagonally. It can also be rotated any way. With that we will start with a few base cases making our job easier later when we will be making a state diagram. When playing I found myself running into many of the same boards. A very common one is the 2x2 square. I have shown below a partial state diagram to show that this is a cold board state.

So this shows that whatever player has to play on this board that they will lose. Bellow is other boards that are in a cold state with a winning strategy. I encourage you to try to show that they are cold boards.

Now that we have these it will clean up our state diagrams because we no longer have to go until the board is completely empty. I will be doing the partial state diagrams in three different varieties. The first will be when the second player only decides to pop one square. The first move of the winning strategy is popping one square in any of the corners. Due to symmetry I only need to show what it looks like from one corner.

Winning Strategy when the 2nd Player Pops One Square.

This shows that whenever the second player pops one square there is a winning strategy. We will now do the same thing for when the second player pops either two or three squares.

The second Player Pops Two Squares.

Second Player Pops Three Squares.

So now we have shown that no matter what the second player does it puts the board in a cold state, this means that at the start of the game the board is in a hot state. From that we have that the first player can always win.

If you ever find yourself playing this game know that if you pop one of the corners then follow the winning strategy you will always win. There may be other winning strategies but I believe the simplest of them is this one.

There is also a much larger version with 28 total squares that can be popped. Maybe one day I will try to find a winning strategy, though I have a feeling it will take a little more time.

Thank you very much for giving this a read! Enjoy the rest of you day.