Solution to Problem 150

Congratulations to this week's winner

Mark Chakoian

Further correct solutions were submitted by Jan Siwanowicz, Denis Borris, Paul Botham.

Although not my intention, this problem has no solution. The problem lies in the restriction that the two numbers be no more than 20. Remove that restriction and there is a solution. The analysis, courtesy of Mark Chakoian, proceeds as follows.

The number that Samantha hears could be from 5 to 39. With but one exception, each of these numbers can be written as either the sum of two primes (in which case Paul would already know the numbers) or the sum of a prime at least 11 and another number (in which case Paul would be able to deduce easily the two numbers). The only exception is 11, namely, 11 = 9 + 2 = 8 + 3 = 7 + 4 = 6 + 5. However, 9 × 2 = 18 = 6 × 3, 8 × 3 = 24 = 6 × 4 = 12 × 2, 7 × 4 = 28 = 14 × 2, and 6 × 5 = 30 = 15 × 2 = 10 × 3. So once Paul hears Samantha say that she knows that he doesn't know, he knows that the factors add to 11, but Samantha appears to have no way to determine which of the above products Paul has.

A complete solution (auf Deutsch) was written by Werner Seischab.

Burkart Venzke reports that the problem has been discussed often in de.rec.denksport. Variations on the theme can be found at Variation 1 and Variation 2.

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