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PUZZLE: There are twelve identically looking balls, all weighing the same but one ball. That ball could either be heavier OR lighter than the rest. You’re given a scale that can ONLY tell you if one side is heavier or lighter or weighs the same compared to the other side. Find a way to determine which ball weighs differently, and if it’s heavier or lighter, in three weighs. SOLUTION: NOTE: Balls that have been eliminated (i.e., they weigh the same as 10 other balls) will be shaded out in gray. Balls that are either heavier OR the same as the other balls are in purple. Balls that are either lighter OR the same as the other balls are in green. Balls in white are unknown. WEIGH #1: 1 2 3 4 5 6 7 8 9 10 11 12 A B If A = B, then go to WEIGH #2a If A > B, then go to WEIGH #2b If A < B, then go to WEIGH #2c WEIGH #2a: What we know: One of balls 9-12 weighs differently from the others. Balls 1-8 all weigh the same. 1 2 3 4 5 6 7 8 9 10 11 12 C D If C = D, then go to Weigh #3a If C > D, then go to Weigh #3b If C < D, then go to Weigh #3c WEIGH #3a: What we know: Balls 1-11 weigh the same. Ball 12 is either heavier or lighter. 1 2 3 4 5 6 7 8 9 10 11 12 E F If E > F, then Ball 12 weighs less than all the other balls. If E < F, then Ball 12 weighs more than all the other balls. Note that E = F is NOT possible, since we already determined Balls 1-11 weigh the same. 1 WEIGH #3b: What we know: Balls 1-8 & 12 all weigh the same. Since 8 & 9 > 10 &11, then either Ball 9 is heavier than the others, or Ball 10 or Ball 11 is lighter than the others. 1 2 3 4 5 6 7 8 9 10 11 12 E F If E = F, then Ball 11 is lighter than all the other balls. If E > F, then Ball 10 is lighter than all the other balls. If E < F, then Ball 9 is heavier than all the other balls. WEIGH #3c: What we know: Balls 1-8 & 12 all weigh the same. Since 8 & 9 < 10 &11, then either Ball 9 is lighter than the others, or Ball 10 or Ball 11 is heavier than the others. 1 2 3 4 5 6 7 8 9 10 11 12 E F If E = F, then Ball 11 is heavier than all the other balls. If E > F, then Ball 9 is lighter than all the others balls. If E < F, then Ball 10 is heavier than all the other balls. WEIGH #2b: What we know: Balls 9-12 all weigh the same. Either one of Balls 1, 2, 3, or 4 is heavier, or one of Balls 5, 6, 7, 8 is lighter. 1 2 3 4 5 6 7 8 9 10 11 12 C D If C = D, then go to WEIGH #3d If C > D, then go to WEIGH #3e If C < D, then go to WEIGH #3f WEIGH #3d: What we know: Balls 3-6 & 8-12 weigh the same. Either 1 or 2 weigh more or 7 weighs less. It can’t be vice versa (ie, 1 or 2 weigh more) (see What we know from WEIGH #2b) 1 2 3 4 5 6 7 8 9 10 11 12 E E F 2 If E = F, then Ball 1 is heavier than all the other balls. If E > F, then Ball 2 is heavier than all the other balls. If E < F, then Ball 7 is lighter than all the other balls. WEIGH #3e: What we know: Balls 1-2, 5-7, 9-12 weigh the same. Either Ball 3 or 4 weigh more, or 8 weighs less. It can’t be vice versa (i.e., 3 or 4 weigh less) (see What we know from WEIGH #2b) 1 2 3 4 5 6 7 8 9 10 11 12 E E F If E = F, then Ball 4 is heavier than all the other balls. If E > F, then Ball 3 is heavier than all the other balls. If E < F, then Ball 8 is lighter than all the other balls. WEIGH #3f: What we know: Balls 1-4, 7-12 weigh the same. Either 5 or 6 weigh less. Note: Ball 8 cannot weigh more (see Weigh #1). It can’t be vice versa (ie, 5 or 6 weigh more) (see What we know from WEIGH #2b) 1 2 3 4 5 6 7 8 9 10 11 12 E F If E = F, then Ball 6 is lighter than all the other balls. If E < F, then Ball 5 is lighter than all the other balls. Note: E > F is NOT possible. WEIGH #2c: What we know: Balls 9-12 all weigh the same. Either one of Balls 1, 2, 3, or 4 is lighter, or one of Balls 5, 6, 7, 8 is heavier. 1 2 3 4 5 6 7 8 9 10 11 12 C D If C = D, then go to WEIGH #3g If C > D, then go to WEIGH #3h If C < D, then go to WEIGH #3i WEIGH #3g: What we know: Balls 3-6 & 8-12 weigh the same. Either 1 or 2 weigh less or 7 weighs more. It can’t be vice versa (i.e., 1 or 2 weigh less) (see What we know from WEIGH #2c) 3 1 2 3 4 5 6 7 8 9 10 11 12 E E F If E = F, then Ball 1 is lighter than all the other balls. If E > F, then Ball 7 is heavier than all the other balls. If E < F, then Ball 2 is lighter than all the other balls. WEIGH #3h: What we know: Balls 1-2, 5-7, 9-12 weigh the same. Either Ball 3 or 4 weigh less, or 8 weighs more. It can’t be vice versa (i.e., 3 or 4 weigh more) (see What we know from WEIGH #2c) 1 2 3 4 5 6 7 8 9 10 11 12 E E F If E = F, then Ball 4 is lighter than all the other balls. If E > F, then Ball 8 is heavier than all the other balls. If E < F, then Ball 3 is lighter than all the other balls. WEIGH #3i: What we know: Balls 1-4, 7-12 weigh the same. Either 5 or 6 weigh more. Note: Ball 8 cannot weigh less (see Weigh #1). It can’t be vice versa (i.e., 5 or 6 weigh less) (see What we know from WEIGH #2c) 1 2 3 4 5 6 7 8 9 10 11 12 E F If E = F, then Ball 6 is heavier than all the other balls. If E > F, then Ball 5 is heavier than all the other balls. Note: E < F is NOT possible. -dennis chen 4

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