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   Another variation to the sum / product riddle
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   Author  Topic: Another variation to the sum / product riddle  (Read 3929 times)
Altamira_64
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Another variation to the sum / product riddle  
« on: Feb 1st, 2013, 6:17am »
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3 logicians A, P and S are again bored to death and looking for ways to drive us crazy Smiley
 
A thinks of two integers >1 and with sum <100.
He then announces their product to P and their sum to S.
 
The following conversation occurs.
 P says "I cannot find these numbers."
 S says "I was sure that you could not find them."
 P says "I knew in advance that you knew I could not find them"
 S says "I don't know the numbers."
 P says "Now I know them!"
 
What are these numbers?
« Last Edit: Feb 2nd, 2013, 1:19am by Altamira_64 » IP Logged
SMQ
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Re: Another variation to the sum / product riddle  
« Reply #1 on: Feb 2nd, 2013, 6:32am »
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P says "I cannot find these numbers." --> the product is not unique.
 
S says "I was sure..." --> all pairs with this sum have non-unique products.
 
P says "I knew in advance..." --> all pairs with this product have a sum such that (all pairs with this sum have non-unique products).
 
By exhaustive search in excel, this leaves 7 candidate pairs:
  6, 17 (sum 23, product 102)
13, 22 (sum 35, product 286)
  3, 34 (sum 37, product 102)
11, 26 (sum 37, product 286)
14, 23 (sum 37, product 322)
  2, 51 (sum 53, product 102)
  7, 46 (sum 53, product 322)

S says "I don't know the numbers." --> the sum is not unique among the candidate pairs
 
This leaves 5 remaining candidates:
  3, 34 (sum 37, product 102)
  2, 51 (sum 53, product 102)
11, 26 (sum 37, product 286)
14, 23 (sum 37, product 322)
  7, 46 (sum 53, product 322)

P says: "Now I know them!" --> the product is unique among the remaining candidates
 
The only remaining possibility is the pair 11, 26
 
--SMQ
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