Some unit cubes are assembled to form a larger cube. Some of the faces of the larger cube are then painted. The cube is taken apart and it is found that 217 of the unit cubes have paint on them. What is the total number of unit cubes?

(In reply to Short solution by Ady TZIDON)

When editing you can yourself permanently erase any previous "Edited On"s that appear, but there will always be one that "the system" adds at the end of any edit. Thus it is always obvious that at least one edit was done, but the exact number of edits done can be suppressed.

The present problem can also be approached as an inclusion/exclusion problem. Take as basic sets the facial cube sets of each face. Then each has n^2 elements, the intersection of any two has n or 0 elements, the intersection of any three has 1 or 0 elements, and the intersection of four or more is always empty. For 3 painted faces, the inclusion/exclsion formula is #A+#B+#C-#AB-#AC-#BC+#ABC so the total number of cubes involved is either 3*n^2 - 2*n + 0 or 3*n^2 - 3*n + 1. The latter proves to be 217 when n=9.

Posted by Richard on 2004-02-02 21:33:18