The chocolate bar game: variants

Remember the chocolate bar game? Today I want to talk about some variants. Recall that the losing positions for the chocolate bar game can be visualized like this:

What if we specify that you win if you end with a 1\times 1 piece of chocolate, instead of losing? Well, that game can be visualized like this:

As you can see, it actually has a lot of the same structure, but it’s as though some rows and columns got shuffled around. Can you explain this?

And here’s what happens if we keep the rules of the original chocolate bar game, but arbitrarily declare that 1 \times 2 is also a losing position (though not 2 \times 1). As I suspected, overall we can still see the same sort of structure of radiating diagonal lines, but with some extra complication thrown in. The result is of course not symmetric.

Zooming out, we can see sequences of dots lying above lines with corresponding dots missing—but not at a fixed distance; the dots seem to come in groups of three that are the same distance away from the line, and evenly spaced from each other; and then the next three are farther away from the line and spaced farther apart, and so on. If true, this is actually very surprising to me; I have no idea where the number 3 would be coming from!

And here’s what we get if we instead arbitrarily declare 13 \times 15 to be a losing position. It looks normal up to a point, but then the extra losing position introduces a disturbance which ripples outwards.

Here is another view of the same game (with losing position 13 \times 15 added), zoomed out twice as far. It looks like it settles into a regular pattern, with the same overall shape as the original game but with some “fuzz”. The fuzz seems to follow some very interesting patterns—zooming even farther out is probably necessary to get a better sense for them!

Interestingly, if we add 13 \times 15 along with its mirror image 15 \times 13, the disturbance becomes more modest.

In this zoomed-out view of the same game, you can see that there will be periodic disturbances, occuring at exponentially increasing distances.

Unlike the chocolate bar game, I have no idea of the right strategy for any of these variants! But I think at least some of them—particularly the reversed form where 1 \times 1 is winning rather than losing—probably have a nice mathematical characterization. I would also really love to see an explanation of why dots in groups of 3 show up when we introduce 1 \times 2 as an extra losing position. Thoughts, comments, etc. welcome! I’m also happy to generate visualizations for other variants you might want to see.

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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