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_ Clue
THE CONTENTS OR THE TEN ENVELOPES (IN DOLLAR BILLS) SHOULD BE AS FOLLOWS: $1, 2, 4, 8, 16, 32, 64, 128, 256, 489. THE FIRST NINE NUMBERS ARE IN GEOMETRICAL PROGRESSION, AND THEIR SUM, DEDUCTED FROM 1,000, GIVES THE CONTENTS OF THE TENTH ENVELOPE THE CORRECT COMBINATION IS 65292. SINCE THE THIRD DIGIT IS THREE LESS THAN THE SECOND, AND THE FOURTH IS FOUR GREATER THAN THE SECOND, THERE ARE ONLY THREE POSSIBLE COMBINATIONS FOR THE SECOND, THIRD AND FOURTH DIGITS. THESE ARE -307-, -418-, AND -529-. WITH THE FIRST DIGIT THREE TIMES THE FIFTH, THE ONLY POSSIBLE COMBINATIONS FOR THE FIRST AND FIFTH DIGITS ARE 0 0,3 1, 6 2, AND 9 3. THE SOLUTION ARISES FROM COMBINING THESE TWO SETS OF POSSIBILITIES, WITH THE ADDED CRITERIA THAT THERE ARE THREE COMBINATIONS OF TWO DIGITS THAT THAT EACH SUM TO 11 THE CHILDREN ARE 1, 6 AND 6 YEARS OLD. THE PRODUCT OF THEIR AGES IS 36, SO NONE OF THEM CAN BE OLDER THAN 36. THE NUMBER 36 HAS TO BE EXPRESSED AS THE PRODUCT OF 3 NUMBERS. THEIR POSSIBLE AGES ARE (THE SUM OF THEIR AGES IS IN BRACKETS): 1, 1, 36 (3938) 1, 2, 18 (21) 1, 3, 12 (16) 1, 4, 9 (14) 1, 6, 6 (13) 2, 2, 9 (13) 2, 3, 6 (11) 3, 3, 4 (10) SINCE CHERYL IS TOM'S NEXT DOOR NEIGHBOUR, TOM KNOWS CHERYL'S HOUSE NUMBER. TOM WOULD KNOW THE CHILDREN'S AGES IN EVERY CASE THAT SUMS UP TO A UNIQUE NUMBER EXCEPT FOR THE SUM OF 13, WHICH HAVE 2 COMBINATIONS OF POSSIBLE AGES. AS A RESULT, TOM WOULD BE CONFUSED AS HE HAS TO PICK BETWEEN THE 2 COMBINATIONS: (1,6,6) AND (2,2,9). CHERYL THEN TELLS TOM ABOUT HER YOUNGEST CHILD WHO LIKES STRAWBERRY MILK WHICH TELLS TOM THAT THERE IS ONLY 1 YOUNGEST CHILD THE SECRET CODE IS 7,4,6,5,8. LET THE NUMBERS = A, B, C, D, E BASED ON 1ST CLUE: E + C = 14 BASED ON 2ND CLUE: A = 2B - 1 BASED ON 3RD CLUE: D = B + 1 BASED ON 4TH CLUE: B + C = 10 BASED ON 5TH CLUE: A + B + C + D + E = 30 ALL NUMBERS CAN BE REPRESENTED BY B EXCEPT FOR E SO WE NEED TO REWRITE EQUATION 1. SUBSTITUTE C = 10 - B FROM EQUATION 4 INTO EQUATION 1. E + (10 - B) = 14 E = B + 4 SUBSTITUTE A,C,D,E IN EQUATION 5 (2B - 1) + B + (10 - B) + (B + 1) + (B + 4) = 30 SOLVING FOR B GIVES 4. USE THE OTHER EQUATIONS TO FIND THE VALUES OF A,C,D AND E