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_ _ C H Clue
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 A MINIMUM OF 3 WEIGHINGS IS REQUIRED. DIVIDE THE 27 BALLS INTO 3 GROUPS OF 9 BALLS EACH. USE THE SCALE TO WEIGH 2 GROUPS. THIS WILL TELL US WHICH GROUP HAS THE HEAVIER BALL. NOW, DIVIDE THE 9 BALLS INTO 3 GROUPS OF 3 BALLS EACH AND WEIGH 2 GROUPS, AND IDENTIFY THE GROUP WHICH HAS THE HEAVIER BALL. FROM THE HEAVIER GROUP, WEIGH 2 BALLS TO DETERMINE THE HEAVIER BALL 8 WEIGHINGS ARE REQUIRED TO FIND OUT THE HEAVY BALL. DIVIDE THE BALLS INTO 3 GROUPS OF 2187 BALLS EACH. PUT 2 GROUPS ON THE SCALE AND DETERMINE WHICH GROUP IS HEAVIER. IF BOTH GROUPS ARE EQUAL IN WEIGHT, THE HEAVIER BALL IS IN THE 3RD GROUP. REPEAT THE PROCESS BY BREAKING THE GROUP WITH THE HEAVIER BALL INTO 3 SMALLER GROUPS OF BALLS AGAIN. FOR THE 2ND ROUND, EACH GROUP WILL HAVE 729 (2187 / 3 ) BALLS EACH. THIS PROCESS HAS TO BE REPEATED 8 TIMES. 1ST WEIGHING - 6561 BALLS ARE DIVIDED INTO 3 GROUPS OF 2187 BALLS EACH. 2ND WEIGHING - 2187 BALLS ARE DIVIDED INTO 3 GROUPS OF 729 BALLS EACH. 3RD WEIGHING - 729 BALLS ARE DIVIDED INTO 3 GROUPS OF 243 BALLS EACH. 4TH WEIGHING - 243 BALLS ARE DIVIDED INTO 3 GROUPS OF 81 BALLS EACH. 5TH WEIGHING - 81 BALLS ARE DIVIDED INTO 3 GROUPS OF 27 BALLS EACH. 6TH WEIGHING - 29 BALLS ARE DIVIDED INTO 3 GROUPS OF 9 BALLS EACH. 7TH WEIGHING - 9 BALLS ARE DIVIDED INTO 3 GROUPS OF 3 BALLS EACH. 8TH WEIGHING - 3 BALLS ARE DIVIDED INTO 3 GROUPS OF 1 BALL EACH TAKE 6 BALLS AND SPLIT THEM INTO 2 GROUPS OF 3 BALLS EACH. USE THE SCALE TO WEIGH THE 2 GROUPS. IF ONE OF THE GROUPS IS HEAVIER THAN THE OTHER GROUP, PICK 2 BALLS FROM THE HEAVIER GROUP AND USE THE SCALE TO WEIGH THEM. IF THEY ARE EQUAL IN WEIGHT, THE OTHER BALL IN THAT GROUP IS THE HEAVY ONE. IF BOTH THE GROUPS ARE EQUAL IN WEIGHT, THEN PICK THE REMAINING 2 BALLS AND USE THE SCALE TO DETERMINE THE HEAVIER BALL