Google researcher, long out of math, cracks devilish problem about sets
Always cool to see unexpected applications of information theory, especially outside of probabilistic contexts.<p>A related toy example that comes to mind is the puzzle where you&#x27;re given 12 metal balls and told that one of them is either heavier or lighter than the rest. You&#x27;re then given a balance&#x2F;scale and instructed to figure out which ball is different and whether it is heavier or lighter using the balance a maximum of three times to weigh different combinations of balls.<p>Part of the reason this is solvable is because there are 12x2=24 different possibilities (or log2(24) = 4.58 bits) for the ball ID&#x2F;weight and 27=3x3x3 possible outcomes (or log2(27) = 4.75 bits) for the weighing process (left heavier, right heavier, or equal in each use). Thus, there&#x27;s a chance the weighing sequence could convey the ball ID&#x2F;weight. If there were 14 balls, however, then you would require log(28) = 4.81 bits of information to specify the ball ID&#x2F;weight, so the puzzle would be unsolvable, since there wouldn&#x27;t be enough information available in the weighing sequence to specify the ball ID&#x2F;weight.