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( W 1 ,
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W _ T H
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G R _ N D D _ _ G H T _ R S
( D 2 ,
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H _ S B _ N D S
( S 1 ,
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( W 2 ,
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( S 1 ,
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M _ T H _ R S
( W 1 , W 2 )
W _ T H
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S _ N S
( S 1 , S 2 )
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( D 1 , D 2 )
W _ T H
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( S 1 , S 2 ) Clue
IF TWO WIDOWS HAD EACH A SON, AND EACH WIDOW MARRIED THE SON OF THE OTHER AND HAD A DAUGHTER BY THE MARRIAGE, ALL THE RELATIONSHIPS WILL BE FOUND TO RESULT. LET'S SAY THE FIRST WIDOW, W1, HAS A SON, S1. THE SECOND WIDOW, W2 HAS A SON, S2. W1 MARRIES S2 AND HAVE A DAUGHTER D1. W2 MARRIES S1 AND HAVE A DAUGHTER D2. SO WE HAVE: 2 GRANDMOTHERS (W1, W2) WITH THEIR 2 GRANDDAUGHTERS (D2, D1) 2 HUSBANDS (S1, S2) WITH THEIR 2 WIVES (W2, W1) 2 FATHERS (S1, S2) WITH THEIR 2 DAUGHTERS (D2, D1) 2 MOTHERS (W1,W2) WITH THEIR 2 SONS (S1,S2) 2 MAIDENS (D1,D2) WITH THEIR 2 MOTHERS (W1,W2) 2 SISTERS (D1,D2) WITH THEIR 2 (HALF) BROTHERS (S1,S2) THE ELDEST IS 9 YEARS OLD AND THE 2 YOUNGER ONES ARE 2 YEARS OLD. LET'S BREAK IT DOWN. THE PRODUCT OF THEIR AGES IS 36. SO THE POSSIBLE CHOICES ARE: 1,1,36 - SUM(1,1,36) = 38 1,6,6 - SUM(1,6,6) = 13 1,2,18 - SUM(1,2,18) = 21 1,3,12 - SUM(1,3,12) = 16 1,4,9 - SUM(1,4,9) = 14 2,2,9 - SUM(2,2,9) = 13 2,3,6 - SUM(2,3,6) = 11 3,3,4 - SUM(3,3,4) = 10 SIX OF THE SUMS ARE UNIQUE, SO IF IT WERE ONE OF THOSE, TOM WOULD HAVE RECOGNISED THE NUMBER ACROSS THE STREET THAT MATCHES AND HE WOULD KNOW THE ANSWER, BUT HE COULD NOT FIGURE OUT THE ANSWER. THIS MEANS THERE ARE TWO OR MORE COMBINATIONS WITH THE SAME SUM. FROM THE CHOICES ABOVE, ONLY TWO OF THEM ARE POSSIBLE NOW. 1,6,6 - SUM(1,6,6) = 13 2,2,9 - SUM(2,2,9) = 13 WHEN TOM HEARD THAT THE ELDEST IS VISITING HIS GRANDFATHER, WE CAN ELIMINATE COMBINATION 1 SINCE THERE ARE TWO ELDEST ONES. THIS LEAVES US WITH ONLY 1 OPTION LEFT, THAT IS 2, 2 AND 9 PAT MUST HAVE PAINTED SIX MORE LAMP POSTS THAN TIM, NO MATTER HOW MANY LAMP POSTS THERE WERE. SUPPOSE THERE WERE 12 LAMP POSTS ON EACH SIDE; THEN PAT PAINTED 15 AND TIM 9. IF THERE WERE 100 LAMP POSTS ON EACH SIDE, PAT PAINTED 103, AND TIM ONLY 97 THE DAUGHTER SHOULD PICK ENVELOPE 1. STATEMENTS 1 AND 2 ARE FALSE, AND THE ONLY TRUE STATEMENT IS STATEMENT 3. IF THE CHECK WAS IN ENVELOPE 1, THAT WOULD MAKE STATEMENT 1 AND STATEMENT 2 FALSE AND STATEMENT 3 WOULD BE THE ONLY TRUE STATEMENT. IF THE CHECK WAS IN ENVELOPE 2, BOTH STATEMENTS 1 AND 2 WOULD BE TRUE. IF THE CHECK WAS IN ENVELOPE 3, BOTH STATEMENTS 1 AND 3 WOULDBE TRUE