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6 Clue
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 FIRST WEIGHING: FOUR AGAINST FOUR SECOND WEIGHING: TWO AGAINST TWO THIRD WEIGHING: ONE AGAINST ONE LET'S NAME THE BALLS 1-12. FIRST WE WEIGH {1,2,3,4} ON THE LEFT AND {5,6,7,8} ON THE RIGHT. THERE ARE THREE SCENARIOS WHICH CAN ARISE FROM THIS. IF THEY BALANCE, THEN WE KNOW 9, 10, 11 OR 12 IS ODD. WEIGH {8, 9} AND {10, 11} (NOTE: 8 IS NOT ODD) IF THEY BALANCE, WE KNOW 12 IS THE ODD ONE. JUST WEIGH IT WITH ANY OTHER BALL AND FIGURE OUT IF IT IS LIGHTER OR HEAVIER. IF {8, 9} IS HEAVIER, THEN EITHER 9 IS HEAVY OR 10 IS LIGHT OR 11 IS LIGHT. WEIGH {10} AND {11}. IF THEY BALANCE, 9 IS ODD (HEAVIER). IF THEY DON'T BALANCE THEN WHICHEVER ONE IS LIGHTER IS ODD (LIGHTER). IF {8, 9} IS LIGHTER, THEN EITHER 9 IS LIGHT OR 10 IS HEAVY OR 11 IS HEAVY. WEIGH {10} AND {11}. IF THEY BALANCE, 9 IS ODD (LIGHTER). IF THEY DON'T BALANCE THEN WHICHEVER ONE IS HEAVIER IS ODD (HEAVIER). IF {1,2,3,4} IS HEAVIER, WE KNOW EITHER ONE OF {1,2,3,4} HEAVIER OR ONE OF {5,6,7,8} IS LIGHTER BUT IT IS GUARANTEED THAT {9,10,11,12} ARE NOT ODD. WEIGH {1,2,5} AND {3,6,9} (NOTE: 9 IS NOT ODD). IF THEY BALANCE, THEN EITHER 4 IS HEAVY OR 7 IS LIGHT OR 8 IS LIGHT. FOLLOWING THE LAST STEP FROM THE PREVIOUS CASE, WE WEIGH {7} AND {8}. IF THEY BALANCE, 4 IS ODD (HEAVIER). IF THEY DON'T BALANCE THEN WHICHEVER ONE IS LIGHTER IS ODD (LIGHTER). IF {1,2,5} IS HEAVIER, THEN EITHER 1 IS HEAVY OR 2 IS HEAVY OR 6 IS LIGHT. WEIGH {1} AND {2}. IF THEY BALANCE, 6 IS ODD (LIGHTER). IF THEY DON'T BALANCE THEN WHICHEVER ONE IS HEAVIER IS ODD (HEAVIER). IF {3,6,9} IS HEAVIER, THEN EITHER 3 IS HEAVY OR 5 IS LIGHT. WEIGH {5} AND {9}. THEY WON'T BALANCE. IF {5} IS LIGHTER, 5 IS ODD (LIGHTER). IF THEY BALANCE, 3 IS ODD (HEAVIER). IF {5,6,7,8} IS HEAVIER, IT IS THE SAME SITUATION AS IF {1,2,3,4} WAS HEAVIER. JUST PERFORM THE SAME STEPS USING 5,6,7 AND 8. WEIGH {5,6,1} AND {7,2,9} (NOTE: 9 IS NOT ODD). IF THEY BALANCE, THEN EITHER 8 IS HEAVY OR 3 IS LIGHT OR 4 IS LIGHT. WE WEIGH {3} AND {4}. IF THEY BALANCE, 8 IS ODD (HEAVIER). IF THEY DON'T BALANCE THEN WHICHEVER ONE IS LIGHTER IS ODD (LIGHTER). IF {5,6,1} IS HEAVIER, THEN EITHER 5 IS HEAVY OR 6 IS HEAVY OR 2 IS LIGHT. WEIGH {5} AND {6}. IF THEY BALANCE, 2 IS ODD (LIGHTER). IF THEY DON'T BALANCE THEN WHICHEVER ONE IS HEAVIER IS ODD (HEAVIER). IF {7,2,9} IS HEAVIER, THEN EITHER 7 IS HEAVY OR 1 IS LIGHT. WEIGH {1} AND {9}. IF THEY BALANCE, 7 IS ODD (HEAVIER). IF THEY DON'T BALANCE THEN 1 IS ODD (LIGHTER). NOTE: THERE ARE OTHER POSSIBLE SOLUTIONS TO THIS PROBLEM AS WELL ZOE'S SMALLEST POSSIBLE NUMBER IS 6. BASED ON THE FIRST STATEMENT OF ALI, IT INDICATES THAT HE HAS NEITHER 1 NOR 9. IF HE HAD EITHER 1 OR 9 THEN HE WOULD KNOW THAT ZOE MUST HAVE A BIGGER OR SMALLER NUMBER. NOW ZOE, BASED ON ALI'S FIRST STATEMENT, KNOWS THAT ALI DOESN'T HAVE 1 OR 9. ZOE'S FIRST STATEMENT INDICATES THAT SHE DOES NOT HAVE 2 OR 8 (NEITHER 1 NOR 9). IF SHE HAD 1, 2, 8 OR 9, THEN SHE COULD HAVE CONCLUDED THAT ALI HAS A BIGGER OR SMALLER NUMBER. NOW ALI KNOWS THAT ZOE DOESN'T HAVE 1, 2, 8 OR 9. ALI'S SECOND STATEMENT INDICATES THAT HE DOES NOT HAVE 3 OR 7 AND ALSO NOT 1, 2, 8 OR 9. ZONE CAN CONCLUDE THAT ALI DOESN'T HAVE 1, 2, 3, 7, 8 OR 9. IN SHORT, ALI MUST HAVE EITHER 4, 5 OR 6. NOW WHEN ZOE SAYS THAT SHE HAS A BIGGER NUMBER THEN IT MUST BE EITHER 6, 7, 8 OR 9 AND ALI HAVING 4 OR 5. ZOE CAN'T SAY CONFIDENTLY THAT SHE HAS A BIGGER NUMBER IF SHE HAD A 4 OR 5, AS IT COULD BE SMALLER THAN WHAT ALI COULD HAVE. SO ZOE'S SMALLEST POSSIBLE NUMBER IS A 6 YOU WOULD ONLY NEED TO TAKE OUT ONE MARBLE BECAUSE WE KNOW THAT ALL OF THE LABELS ARE INCORRECT. SO YOU PULL ONE MARBLE OUT OF THE BOX LABELED "MIXED." IF RED COMES OUT, YOU KNOW THAT HAS TO BE THE ALL-RED BOX, SO YOU PUT THE RED LABEL ON IT. THE BOX LABELLED "BLUE" MUST THEN BE LABELLED "MIXED" BECAUSE YOU KNOW IT IS ALSO LABELED INCORRECTLY, AND THEREFORE CAN'T BE BLUE. YOU WOULD LABEL THE LAST BOX "BLUE" BECAUSE THAT IS THE ONLY COLOR/BOX COMBO LEFT. IF THE FIRST MARBLE YOU PULLED OUT FROM THE BOX LABELED "MIXED" IS A BLUE MARBLE, THEN YOU SOLVE THE PROBLEM IN THE SAME GENERAL WAY