There is a strange circular disk on a table. It has twelve congruent sectors, each of which is either opaque or transparent.

An identical disk is placed on top of it, and spun around. The table underneath the disks is brightly colored, so that every 30 degrees of rotation, you can count the number of regions out of 12 that are transparent. Those numbers are 2, 3, 4, 4, 0, and 4 for the first half of a revolution. Without any more information, can you figure out exactly what a single disk looks like?

There must be 6 transparent sectors. Any more would prevent the possibility of a 0 and any fewer could not allow a such a pattern: Suppose there were 5 transparent sectors, in the part of the pattern which alternates 4, 0, 4 we would necessarily imagine that at most one pair of transparent sectors could be adjacent, but this is clearly not possible when considering that the pattern is preceded by a 4.

Posted by Eric on 2005-12-03 16:36:02