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Guessing Two Consecutive IntegersAs every one knows, there is nothing less compatible with a mathematician's attitude than passive watching. Mathematics is not a spectator's sport. As an apparent exception, here is a problem whose solution at first glance seems impossible, but then shown to exist and consisting at that, for the most part, in plain watching.
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Copyright © 1996-2008 Alexander Bogomolny
SolutionThere is no doubt that in the absence of the clock, there is no way that either A or B would be able to arrive at the other's number. If one has k the other may have either But wait. There is at least one case when a fellow can arrive at the solution in a logical way. If, say, A has been assigned The next simplest case is when, say, A has 2 and B has 3. A may argue then that B has either 1 or 3 and, knowing B for a perfect logician, he would expect an announcement of a solution by B after the first strike should the latter hold integer 1. If B misses the first strike, A will be in a position to conclude that B is holding 3 and not 1 and will be able to make an announcement to that effect after the second strike. Let's summarize: For the distribution 1/2 the owner of the lower number is able to make an announcement after the first strike. For the distribution 2/3 the owner of the lower number is able to make an announcement after the second strike. In both cases this is just what is needed. Looks like we got a good case to evoke mathematical induction. Let Sk be the statement we need to prove, viz., that if the lower number is k, its owner will be able to make the announcement after the kth strike. We just saw that S1 and S2 are indeed true. Assume so is Sm, i.e., assume that for some m, if the assigned numbers are m and Looks like everything falls nicely into its place. Still, there is one question that keeps bugging me: what is exactly the function of the clock? At first sight, it may be called "A Counter of Missed Opportunities" but this would not be quite correct. If the numbers are, say, 30 and 31, the lack of announcement after the first strikes provides no information at all as both A and B know that neither holds, say, 1 or 2, and hence do not expect any announcement after the first strikes. Indeed, the only strike that matters is the last one. Both participants are reduced to waiting for the final strike - sort of unmathematical attitude. To mitigate the grievances observe that this only became possible, albeit regretfully necessary, after a rigorous mathematical analysis.
Copyright © 1996-2008 Alexander Bogomolny |
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