Article 596 of sci.physics: Path: puukko!santra!tut!enea!mcvax!uunet!husc6!uwvax!oddjob!ncar!gatech!mcnc!decvax!decwrl!pyramid!ctnews!andrew!TS0014%OHSTVMA.BITNET@CUNYVM.CUNY.EDU From: TS0014%OHSTVMA.BITNET@CUNYVM.CUNY.EDU Newsgroups: sci.physics Subject: Re: Mathematical Puzzle] Message-ID: 903@sri-arpa.ARPA Date: 21 Mar 88 18:28:19 GMT Lines: 21
From: Joe Damico TS0014%OHSTVMA.BITNET@CUNYVM.CUNY.EDU
Assuming the integers must be "different", it follows that: sum 3 4 5 6 7 8 9
possible 1,2 1,3 1,4 1,5 1,6 1,7 1,8 pairs 2,3 2,4 2,5 2,6 2,7
3,4 3,5 3,6
4,5
If S doesn't know, then sum>4. If S knows P doesn't know, then sum>6. (IF sum=5 then numbers could be 1 and 4, and so P could know the numbers) (IF sum=6 then numbers could be 1 and 5, again, P could know the numbers) SO the numbers could be 1 and 6. P knows the product is 6, but doesn't know whether the factors are (2,3) or (1,6) By saying "I know that P doesn't know", S informs P that the sum is not 5. P says "Now, I know" But, by similar argument, the numbers could be 1 and 8.
Correct me if I'm wrong, but I don't think the problem has a unique solution →Joe Damico
Article 599 of sci.physics: Path: puukko!santra!tut!enea!mcvax!uunet!husc6!mailrus!nrl-cmf!ames!ucsd!nosc!cod!stewart From: stewart@cod.NOSC.MIL (Stephen E. Stewart) Newsgroups: sci.physics Subject: Re: Mathematical Puzzle] Message-ID: 1039@cod.NOSC.MIL Date: 22 Mar 88 23:53:07 GMT References: 898@sri-arpa.ARPA 5818@watdragon.waterloo.edu Reply-To: stewart@cod.nosc.mil.UUCP (Stephen E. Stewart) Organization: Naval Ocean Systems Center, San Diego Lines: 41
In article 5818@watdragon.waterloo.edu bpdickson@trillium.waterloo.edu (Brian P. Dickson) writes:
In article 898@sri-arpa.ARPA Richard Pavelle RP%OZ.AI.MIT.EDU@XX.LCS.MIT.EDU
writes:
P: I don't know what the numbers are.
S: I knew you didn't. Neither do I.
P: Oh! Now I know.
S: Oh! So do I.
What are the two integers?
1 and 4
1⇒product not prime or 1
2a⇒sum odd
2b⇒sum > 3
3⇒product is product of 2 primes since only two ways of getting product
4⇒sum < 7 since only 2 ways of getting sum
I submit that there are exactly 5 possible solutions. The inferences which Brian made from 3 and 4 above are not completely rigorous. More than two ways of getting the product are allowed as long as all but one are eliminated by the requirement that the sum be odd. Any product of two or more primes will be odd unless one (or more) of them is 2. Thus, unless a 2 is involved, the sum of 1 plus the product and the sum of any two numbers derived by taking subproducts will always be even and 2a would not be satisfied. So, at least one of the prime factors must be a 2. In this case, the sum of 1 plus the product will be odd. But, unless all of the prime factors are twos, at least one pair of numbers derived by taking subproducts of the prime factors will also give an odd sum and P could not determine the numbers from 2a. Only when all of the prime factors of the product are twos will all possible pairs of numbers resulting in the product have an even sum except one, namely 1 and the product itself. Thus, 2a gives P the answer. From the knowledge that P then knows the two numbers, S will be able to deduce from the foregoing arguments that the only possibilities are: 1&4, 1&8, 1&16, 1&32, and 1&64. Knowing the sum, he can determine which one it is and announce that he knows also.
Steve Stewart