Empty a cubo

1. In the specific case, a long sequence is not necessary.
2. Try to fold the middle cube.
3. In general, think of a version of Euclid's algorithm: reducing one cube using the other two.

In the specific case, one movement is enough. We start from 12, 7, 3. If we double the cube of 7, we need to add 7 marbles to it. We can take them like this:

• 4 of the cube of 12
• 3 from the cube of 3 So we get 8, 14, 0 and there is already an empty cube. In general, it can also always be done. The idea is to order the cubes from smallest to largest, such as a, b and c, and use the movements to reduce the median cube b, in a way similar to Euclid's algorithm. We write b = qa + r, with r between 0 and a-1. Through a succession of foldings of the small cube, it can be achieved that the medium cube loses exactly qa marbles and is reduced to r, which is already less than a. Then the cubes are rearranged and the operation is repeated. At each stage, the median cube strictly decreases. Since it cannot decrease indefinitely without reaching a smaller non-negative integer, the process necessarily ends when one of the buckets reaches 0.