Math Puzzle No. 4: The Hungry Mouse

By Anonymous

Imagine a block of cheese has been cut into thirds along each face. A friendly neighborhood mouse comes along and starts eating one of the corners. The mouse will eat the entire block of cheese, always progressing to an adjacent cube using only up, down, or cardinal directions (North, South, East, West;  sorry, no diagonals!). 

Challenge: Can you find a path following these rules so that our mouse friend finishes by eating the center cube of cheese last?

Hint: A diagram showing the block cut into cubes -- and another showing the faces made up of cubes --- may help you visualize the problem.

Update: 1/25/21: read below for the solution.

Upon initial inspection, we would likely begin solving a problem like this by taking the grid and trying to find a route for the mouse to eat all of the cheese, somehow ending on that sweet, delicious, final center cube. It is so tempting to just sit and spend a few hours mindlessly drawing grids and different routes through them. However, after a few hours of near successes, (and our friends looking at us like we're going a wee bit mad), we might start to suspect that such a path is impossible! But now we have a new problem! Short of writing down every imaginable route, how would we prove that our poor mouse friend is out of luck and that she'll just need to eat her cheese out of order and maybe she should see a professional to help manage her OCD tendencies.

It turns out, that we can solve this problem for her in just a few sentences by introducing a bit of artificial information! Let's start by labeling the corners where the mouse could start, and then labeling only those squares the mouse could get to after 2 moves with dots, as below:
Now, we can see that there is a total of 27 cubes of cheese for the mouse to eat: 14 with a dot, the other 13 of them dotless. According to our mouse's preference, she will always begin on a dotted cube, and move back and forth between dotless and dotted. After 26 cubes of cheese have been consumed, our dear friend mouse will have eaten 13 with dots, and 13 without. Oh no! We are always then left with a dotted cube, and the center cube was dotless! Sadly, this means we must quietly take our good friend mouse (with whom we have spent many hours trying to solve this problem) aside and let her know she needs a new approach. Has she considered taking up peanut butter?




The previous Math Puzzle and its Solution are available here: https://mathcep.blogspot.com/2020/11/math-puzzle-no-3-ten-divisibilities.html

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