PROBLEM OF THE MONTH

FEBRUARY 2002


 






A cube of cheese of size 3x3x3 is divided into 27 small cubes of size 1x1x1. A mouse eats one small cube each day and an adjacent small cube (that is sharing a face) the next day. Can the mouse eat the center small cube on the last day ?

There are several possible solutions to this problem including a ... "brute force" approach of writing computer code that would enumerate and test all possible scenarios!

Here is one way to model and solve the problem:

The are altogether $ 27$ small cubes of size $ 1\times1\times1$ and one may divide them in two groups:

  1. The first group will include the $ 8$ small cubes which are situated at the corners of the big cheese cube, together with the $ 6$ small cubes located at the centers of each of the faces. Thus a total of $ 14$ cubes.
  2. The second group consists of all the remaining small cubes, thus a total of $ 13$.

Observe now that each time the mouse eats an allowed small cube of cheese, it has to pass from a cube in the first group to one in the second group or vice-versa. On the other hand the center cube belongs to the second group (of only $ 13$ small cubes). So if there would be a way such that the mouse could eat the center small cube on the last day, then it should have eaten a small cube belonging to the second group also on days $ 1,3,5,\ldots,25$, which is impossible since the are only $ 13$ cubes in that group.