al@mot.UUCP (Al Filipski) (04/13/85)
Can someone help me reconstruct this puzzle? I have never seen it
in print; the fellow I heard it from (a couple of years ago)
cannot remember it exactly either, but thinks it might have come
from some magazine for mathematics teachers. (In a sense this is
a second order puzzle or metapuzzle, since the puzzle is to reconstruct
the puzzle)
It goes something like this: It involves two logicians and a third
party. The third party chooses two integers (I don't remember if the
range of the integers was restricted) and tells the first
logician the sum of the numbers and the second logician the product
of the numbers. Say the logicians' names are Athol and Elba. Then
the following conversation ensues:
Athol: I can't tell what the numbers are.
Elba: I can't tell what the numbers are.
Athol: Now I know what the numbers are.
Elba: Now I know what the numbers are.
From this it is possible to deduce the two numbers.
I don't remember which of the above logicians was the one that knew
the sum. (No remarks about my not knowing my Athol from my Elba).
Does anyone know the source or correct version of the above or any
similar puzzle?
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Alan Filipski, UNIX group, Motorola Microsystems, Tempe, AZ U.S.A
{allegra|ihnp4}!sftig!mot!al OR {seismo|ihnp4}!ut-sally!oakhill!mot!al
ucbvax!arizona!asuvax!mot!al
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Orangutans are skeptical of changes in their cages.js2j@mhuxt.UUCP (sonntag) (04/17/85)
> It goes something like this: It involves two logicians and a third > party. The third party chooses two integers (I don't remember if the > range of the integers was restricted) and tells the first > logician the sum of the numbers and the second logician the product > of the numbers. Say the logicians' names are Athol and Elba. Then > the following conversation ensues: > > Athol: I can't tell what the numbers are. > Elba: I can't tell what the numbers are. > Athol: Now I know what the numbers are. > Elba: Now I know what the numbers are. > > -------------------------------- > Alan Filipski, UNIX group, Motorola Microsystems, Tempe, AZ U.S.A I think I've found a way to work this puzzle if we restrict the range of the two integers to 1 through 9 and say that Athol knows the product and Elga knows the sum. It may not be necessary to restrict the range to 9, but it makes the problem a little easier. Here goes: Elba, having heard Athol say that he couldn't tell what the numbers are, knows that the pair must be in the set of pairs which produce a non- unique product. He could list them exhaustively: Product 1 pair 2nd pair Sums 4 2 2 4 1 4,5 18 3 6 9 2 9,11 12 3 4 6 2 7,8 8 2 4 8 1 6,9 6 2 3 6 1 6,7 9 3 3 9 1 6,10 36 4 9 6 6 13,12 24 4 6 8 3 10,11 Unluckily, however, the sum that Elba knows is not one of the unique sums on that list like 4,5,13,12, or 8, so Elba can't tell which pair it is and tells Athol so. Athol, knowing the product, has narrowed the list of cantidate pairs down to two. Since the knowlege that the pair has a non-unique sum is enough to let him know which pair it was, we can be sure that his list of cantidate pairs has only a single pair which has a non-unique sum and a pair with a unique sum. The only such set of cantidate pairs in the list above is (3 4) and (6 2). Since 3+4=7 (non-unique) and 6+2=8 (unique), the pair must be (6 2). Athol, being a logician, figures all of this out in a flash, and states that he knows what the pair of numbers is. As soon as Elba finds out that Athol has figured it out with only the information that Elba couldn't figure it out earlier, Elba follows the above line of reasoning and figures it out too. Clever devils, those logicians! -- Jeff Sonntag ihnp4!mhuxt!js2j "I never met a man I didn't like."- M. Trudeau