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Going around the disk, the twelve regions are the following (T means transparent and O means opaque): TOTOTTOOTTOO. The disk that is on top is identical, but it has been flipped over to the opposite side.
Method (in brief):
Most would probably first consider the possibility that both disks are face up (neither has been flipped). If this were the case, the numbers would follow a certain pattern. At 0 degrees, the number is equal to the number of transparent regions. At 180 degrees, the number is even. All the numbers inbetween show symmetry, that is, the number for 150 degrees is equal to that of 210 degrees, and 120 matches 240, etc. The sum of all the number should be N^2, where N is the number of transparent regions.
Looking at the given numbers, we can deduce that the number of transparent regions is 6, and that 180 degrees corresponds with either the last 4 or the 0. It cannot be the 2, since this would result in a total of 38 rather than 36. We may try to assume that either the 4 or 0 correspond to 180 degrees, but neither assumption will be successful. I'm skipping details, but you may see comments for deeper explanations.
If we assume that the top disk has been flipped over, then the number patterns are quite different. The sum of all the numbers is still N^2. Each number corresponds to a particular reflection permutation. The reflections alternate between 1. reflection over one of the 6 diameters that separate the regions, and 2. reflection over the centers of a pair of regions. The former must always show even numbers and the latter may show even or odd. With this information, one may systematically find the solution (and this is where most of the work is), but I will not show any further details here. |
