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s16 - Now we deal with the case in which the scale does not...

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Solution to Problem 16 Congratulations to this week’s winners Ray Kremer and Mike Fitzpatrick A correct solution was also received from Dr. David Quigg. No correct solutions were received for the general problem. It will remain open until the end of the semester, and the prize for its solution will go up to $5. Denote the twelve balls by the integers 1 to 12. In the first weighing, place balls 5 to 8 on one side and 9 to 12 on the other side of the scale. If the scale balances, the bad ball must be among balls 1 to 4. In this case you put balls 1, 2, 3 on one side and three good balls, say 10, 11, 12, on the other. If they balance, ball 4 must be bad and one more weighing will determine whether it’s light or heavy. So we can suppose they don’t balance, and, for sake of argument, one of balls 1, 2, 3 is heavy. (The case in which one of the balls is light is treated similarly.) Now weigh ball 1 against ball 2. If they balance, then ball 3 is bad and it’s heavy. If they don’t balance, the ball weighing heavy is the bad one.
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Unformatted text preview: Now we deal with the case in which the scale does not balance. I’ll assume the balls are labeled so that one of the balls 9, 10, 11, 12 is heavy or one of the balls 5, 6, 7, 8 is light. Place balls 5, 9, 10 on one side and balls 6, 11, 12 on the other. If the scale balances, ball 7 or 8 must be bad and it must be light. Weighing them against each other will figure it out. So we can suppose they don’t balance, and that the balls are labeled so that the pan with the balls 6, 11, 12 is heavier than the pan with the balls 5, 9, 10. This allows us to conclude that one of ball 11 or 12 is heavy or ball 5 is light -- this follows because the bad ball cannot be on the light side of the scale for the first weighing and the heavy side of the scale for the second weighing. For the last weighing, put balls 5, 11 on one side and two good balls, say 1,2, on the other. Knowing that 5 is light or that 11 or 12 is heavy will allow you to conculde which ball is bad....
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