Play with a friend and alternate turns Each of your turns consists of choosing or CHOMPing a square and along with other squares to the right and above of it. The lower left square is, however, POISONED so the player who is forced to eat it loses
Try this game with a friend... or better yet, a computer:
You may notice the computer link above only has a 4 by 7 grid of chocolate squares... so before reading this post, why not try some other sizes with yourself or your friends. Can you figure out some guaranteed winning strategies for small boards of 2 by 2 or 2 by 4? How about trying a massive board of 13 by 29?!
It will not take long for you to realise that finding a winning strategy is quite difficult for these boards, so hard that you may question whether they exist? Could we, as mathematicians, prove that they exist or not? Finding them is secondary, let us begin by actually trying to show their existence.
Strategy Stealing Argument
In mathematics, there is a common field of study known as game theory that analyses these kinds of situations and "games" to figure out winning strategies, optimal strategies, maximum number of moves, etc. Over time, this field has developed various arguments and theorems that have significantly simplified the process of debunking a problem. One of these is strategy-stealing arguments.
The strategy-stealing argument is a general concept that suggests that in many games, the second player can't have a guaranteed way to win. The idea behind this argument is that if the second player had a foolproof strategy for winning, we could adapt it to create a winning strategy for the first player. This doesn't make sense because both players can't have surefire ways to win. Therefore, in such games, either the first player has a winning strategy, or the game ends in a tie.
This strategy-stealing argument applies to games where both players have similar moves available to them and where making an extra move doesn't put you at a disadvantage. It's used to conclude that the first player typically has an advantage. However, it doesn't provide specific details about what those winning strategies might be.
In Practice...
If this sounds confusing, let us apply it to our game! This is our board:
and as the first player I will take the top right square.
Now, the computer responded with taking out the square right beside this.
Think about what this means. It means the computer believes the winning strategy is to take out the top right two squares. So, ideally, the position the computer has put me in should make me lose, right?
So, now I will start a new game....
Instead of taking the top right square, I will take out the top two squares.
Looks familiar? This is the exact position the computer had put me in thinking it would beat me... so without knowing the winning strategy itself, I have just put the computer in a losing position just like I was! This argument can be tricky to wrap your head around, read this section as many times as you can before you have that eureka moment of... Oh, we have just proved Player 1 can always win.
In other words, the above argument is saying : Player A will play a dummy move to check what Player B's strategy is. Then, Player A just copies that strategy from Move 1 so that Player B can never use it. In doing so, Player A can always win because any winning strategy of Player B can be used by Player A earlier (since he/she plays first move).
This may seem trivial to some but extremely helpful others. However, the caveat of this argument is that we cannot actually find the winning strategy... we know it exists though.
Winning some Cases
Square Boards (n by n)
Remove the square diagonally top right to the poison square (hence leaving only one row and one column)
Mimic the number of squares the opponent removes on the opposite side (if they remove one square from the column, remove one from the row. if they remove three from the row, remove three from the column)
Eventually, the only square remaining would be the poison square and a row or a column of squares... and if you have done this right, it should also be your turn
Now, you can remove everything but the poison square and force the loss for the opponent
Short rectangles (2 by n)
Remove the square on the top right of the rectangle (hence only one square is removed)
The opponent can either remove a set of squares from the top row or the bottom row
If it is the bottom row, you are back to step 1 and just repeat
If it is the top row, you should remove a set of squares from the bottom row. Remove them from one right to the one they did (i.e, if they removed top row squares of Column 4 from the left, then remove bottom row squares of Column 5 from the left)
Now, you can repeat steps 2, 3, and 4 until you reach a 2 by 1 rectangle where it is your turn
Simply removing the top square, forces the opponent to take the poison and lose!
Cheap tricks
Open the CHOMP! Game on two separate tabs
Chomp the top right square on the first tab
Look at the computer's response on the first tab and mimic it on the second tab
Now, copy the response of the computer on the second tab and mimic it on the first
Repeat steps 3 and 4
Eventually, you will have lost your game on the first tab, but won on the second tab!!!
Bibliography and Suggested Further Reading
Ekhad, Shalosh B., and Doron Zeilberger. “All the Winning Bites for a by B Chomp for a and B up to 14 and Two Computational Challenges.” Sites.math.rutgers.edu, 11 Aug. 2018, sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/chompc.html.
Elduque, Eva. “The Game of Chomp.” University of Wisconsin Math Department Wiki, wiki.math.wisc.edu/images/Chomp_Sol.pdf.
Ferguson, Thomas S. “Chomp.” UCLA Mathematics, www.math.ucla.edu/~tom/Games/chomp.html.
Shan, Kaze. “Chomp.” YouTube, 29 Oct. 2014, www.youtube.com/watch?v=nFI-KbeOHRo.