G R _ N D F _ T H _ R
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6 6
_ N D
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6 .
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6 1 ,
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X
6 1 )
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6 2 ,
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6 6 ,
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6
( 6 6
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6
X
1 1 ) Clue
THE PRINCESS IS 40 YEARS OLD AND THE PRINCE IS 30 YEARS OLD (THERE ARE MULTIPLE SOLUTIONS). LET CURRENT AGE OF PRINCE = X CURRENT AGE OF PRINCESS = Y YEARS IN FUTURE = P YEARS IN PAST = Q EQUATION 1: Y = X + P EQUATION 2: Y + P = 2(X - Q) EQUATION 3: Y - Q = 1/2(Y + X) WE HAVE 3 EQUATIONS, WITH 4 UNKNOWNS. THIS IMPLIES THAT THERE ARE MULTIPLE SOLUTIONS. LET'S SET EVERYTHING EQUAL TO 0 FIRST (AND MULTIPLY EQUATION 3 BY TWO TO MAKE IT EASIER). Y - X - P = 0 Y + P + 2Q - 2X = 0 Y - 2Q - X = 0 WE'RE LOOKING FOR THE AGE OF THE PRINCE AND AGE OF THE PRINCESS (X AND Y), SO WE WANT P AND Q TO EQUAL ZERO. INSTEAD OF USING A MATRIX, I TOOK ADVANTAGE OF THE CONVENIENT SITUATION (LOOK AT THE VALUES OF P AND Q IN THESE 3 EQUATIONS - WHAT HAPPENS WHEN YOU ADD THEM?) AND ADDED ALL THE EQUATIONS TOGETHER. THEREFORE: 3Y - 4X = 0 EITHER: (1) Y = 4/3X OR (2) X = 3/4Y WHICH IN ENGLISH MEANS: THE PRINCE IS 3/4 THE AGE OF THE PRINCESS 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 GRANDFATHER WAS 66 AND GRANDSON WAS 6. WHEN GRANDFATHER WAS 61, GRANSON WAS 1 (61 = 1 X 61) WHEN GRANDFATHER WAS 62, GRANSON WAS 2 (62 = 2 X 31) WHEN GRANDFATHER WAS 63, GRANSON WAS 3 (63 = 3 X 21) WHEN GRANDFATHER WAS 64, GRANSON WAS 4 (64 = 4 X 16) WHEN GRANDFATHER WAS 65, GRANSON WAS 5 (65 = 5 X 13) WHEN GRANDFATHER WAS 66, GRANSON WAS 6 (66 = 6 X 11) WE CONVERSED ON JANUARY 1 AND HER BIRTHDAY WAS ON DECEMBER 31. SO, THE DAY BEFORE YESTERDAY ON DEC. 30TH SHE WAS 22 AND HE TURNED 23 ON DEC. 31. SO HER NEXT BIRTHDAY, WHEN SHE TURNS 24, WOULD BE DEC. 31 OF THE SAME YEAR THE QUESTION WAS ASKED. HOWEVER, NEXT YEARS BIRTHDAY WOULD BE THE FOLLOWING YEAR ON DEC. 31, WHEN SHE WOULD BE 25