Guessing Two Consecutive Integers

As 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.

Two people, say A and B, are assigned consecutive positive integers. They are each informed of their own integer and made aware that the two integers are consecutive. The task for them is to guess the other fellow's integer.

The fellows sit in the same room but are forbidden to communicate in any way. Aside of the chairs they sit on, the only other device in the room is a wall clock that strikes every hour. A and B are instructed to announce their solution as soon as they get one but only immediately after a clock strike.

Under the assumption that the two are consummate logicians and both are aware of and rely on each other's power of logical thinking prove that, if the two numbers are k and k +1, the owner of the lower number will announce his solution after the k^th strike of the clock.

Solution

References

J. Havil, Impossible?, Princeton University Press, 2008, pp. 9-10

Solution

There 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 k -1 or k +1 and, short of just guessing, the owner of k has no information to base his thinking on.

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 k = 1, B would necessarily be assigned 2, since the assigned numbers are said to be positive. So the owner of 1 may immediately announce that the other's number is 2 and will do that after the first clock's strike. This is a beginning!

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 S_k 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 k^th strike. We just saw that S₁ and S₂ are indeed true. Assume so is S_m, i.e., assume that for some m, if the assigned numbers are m and m +1, then the owner of m is able to announce a solution after the strike #m. Assume now that the assigned numbers are m +1 and m +2. The owner of the lower number (i.e., m +1) may argue that the other fellow holds either m or m +2. In the former case, he would expect an announcement after the m^th strike and seeing it missed, will have sufficient grounds to conclude that the other number is m +2 leading to an announcement at the strike # m +1.

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.

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