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#11
07-23-2005, 10:18 PM
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Re: Two numbers and two logicians
[ QUOTE ]
<font color="white"> multiplying all valid products, the only standalone sum was 17. All other eligible sums had multiple combinations. My answer is 4 and 13. </font> [/ QUOTE ] You didn't even use the conversation. 
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#13
07-24-2005, 01:01 AM
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Re: Two numbers and two logicians
Begin
<font color="white"> I'll refer to Tarski's and Godel's statements as T1,G1,T2,G2 Pairs of primes are excluded by T1. Also excluded are any two numbers over 50. Also exluded are pairs where any prime factor of one number is greater than 99 divided by the other number. Also excluded are the pairs (2,4),(3,9),(5,25),(7,49) G1 Sums exclude those numbers which can be written as the sum of two numbers excluded by T1. T2 pairs are those not excluded by T1, such that exactly one combination of primes in the product yields a sum not excluded by G1. Finally, a G2 sum is one not excluded by G1, such that exactly one pair yielding the sum is a T2 pair. List primes for reference: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61, 67,71,73,79,83,89,97 Listing G1 numbers. Notice 57=53+4, 59=53+6 etc. 11,17,23,27,29,35,37,41,47,51,53 11=9+2 is a T2 pair since 3+6 is not G1 11=8+3 is a T2 pair since 4+6 and 2+12 are not G1 So 11 does not produce a unique T2 pair and thus is not G2. 17=15+2 is not T2 since 5+6 is G1 17=14+3 is not T2 since 2+21 is G1 17=13+4 is T2 since 26+2 is not G1 17=12+5 is not T2 since 3+20 is G1 17=11+6 is not T2 since 33+2 is G1 17=10+7 is not T2 since 2+35 is G1 17=9+8 is not T2 since 3+24 is G1 So 17 produces the unique T2 pair (13,4). We could exhastively show the remaining G1 numbers do not produce unique T2 pairs but this is unnecessary because the logicians must have already done this in order to make the G2 statement possible. Therefore the numbers are 4 and 13. </font> End I'm still thinking about those 12 marbles though. PairTheBoard
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#15
07-24-2005, 01:42 AM
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Re: Two numbers and two logicians
Addendum
<font color="white"> I should probably correct my last statement to say, it's not the G2 statement that allows us to dismiss the other G1 numbers as candidates but the fact that the puzzle master has told us we can reach the conclusion based on the information given. If we checked the other G1 numbers and found another that produced a unique T2 pair, then we could not come to a conclusion. Since the puzzle says we can come to a conclusion we know that if we did check the other G1 candidates none would produce a unique T2 pair. </font> End PairTheBoard
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#17
07-24-2005, 08:27 AM
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Re: Two numbers and two logicians
<font color="white"> I agree with 4 and 13 because you can use a sieve of Eratosthenes-type method to reduce the possibilities for the sum of n+m to:
11,17,23,27,29,35,37,41,47,51,53,57,59,65,67,71,77 ,79,83,89,93,95,97 after the first two parts of the conversation. If the sum were anything else, it would have been impossible for Godel to make the statement he did. Then starting with a sum of 11 you can check the possible pairs of numbers to see if the rest of the conversation makes sense... With 11 it can't happen because the possible numbers are: 2,9 3,8 4,7 5,6 Here Godel could not possibly have known whether (2,9) or (4,7) were correct because the first three lines of the conversation could have taken place in either case. So a total of 11 must be ruled out. Next is 17... Here the possibilities are: 2,15 3,14 4,13 5,12 6,11 7,10 8,9 Here the only pair for which Tarski could have stated what he did is (4,13). For all others there are at least 2 sums that could have been possible from Godel's statement. For example, in the case of (2,15), the product of 30 could be arrived at with (2,15) or (5,6), each of which would have produced a sum on the above list making it impossible for Tarski to know the sum. The other possiblilities can be ruled out accordingly. We now have one possible solution. To prove that there are no others we'd have to examine all the sums above 17 which is a tedious task indeed, but we can at least rest assured that either we have found the solution or the question has no unique solution. </font>
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