Enigma 1639: Square clocks
11 December 2011
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From New Scientist #2805, 26th March 2011 [link]
In, say, 5-clock arithmetic, counting runs from 1 to 5, but starts again at 1 when 6 is reached. So, in 5-clock arithmetic, 3 + 4 = 2 and so on. Anne, Brian and Cate have been investigating x-clock arithmetic, where x is 2 to 15 inclusive. They have each discovered a two-digit square which, expressed in their three different clocks, gives the same number y.
If I told you the value of y, you could tell me the value of the square and the three values of x they were considering.
NOTE: As stated the problem does not specify what the required solution is. There are three distinct solutions, and the question has no way of distinguishing which is the “right” one.
When the intended solution was published in the magazine it became clear that in order to arrive at that solution you needed to use standard modulo arithmetic, and not the variation explicitly given in the question. A clarification was later published, accepting the three solutions as the answer.