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« The emeralds of the Maharaja »

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The 12 emeralds The Maharaja of Saharanpur has 11 daughters. On the 12th birthday of its government, he decides to offer one emerald to each of his daughters. He owns 12 gems, each of them identically tailored and all weighting 12-carat except one, of a slightly different weight. He ignores which one is the different one and does not even know if it is heavier or lighter than the others. Nevertheless, he wants to be totally equal to all of his daughters. He owns a very old yet very precise balance which can support only three weightings and compare two trays at a time. How can he identify the different emerald?

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The proposed solution also allows to clarify if the different emerald is lighter or heavier

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1 2 3 4 5 6 7 8 9 10 Case 1 ?OKKO 11 Case 1.1 121 Case 1.1.1 11 Weighting 1Weighting 2Weighting 3 or 11 12 12 is different and the balance tells us how (light or heavy) 11 is too light 101 Case 1.2 1 9 10 11 9 109 Case 1.1.2 10 Case 1.1.3 9 is too heavy 9 Case 1.1.4 10 is too heavy And symmetrical cases with balance bending on the right if 11 is too heavy or 9-10 too light 109 Next weighting will be done using 1 of the weighted emerald (that is obviously OK) with 3 of the remaining emeralds

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11 1 2 3 4 5 6 7 8 9 10 11 12 Case 2 ?OKKO 9 10 Case 2.1 12 8 4 Case 2.1.1 12 8 4 1 2 3 4 5 6 7 8 11 9 10 Case 2.2 1 2 3 4 5 6 7 8 Case 2.1.2 6 7 5 Case 2.2.1 6 7 5 Case 2.2.2a 6 7 5 Case 2.2.2b 8 is too light 4 is too heavy 7 is too light 5 is too light6 is too light Weighting 1Weighting 2Weighting 3 Next weighting will be done by shifting 3 out of the 4 emeralds on the other side and replacing them by 3 of the remaining emeralds (that are OK)

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11 1 2 3 4 5 6 7 8 9 10 11 12 Case 2 ?OKKO 9 10 Case 2.3 84 3 5 6 7 2 2 3 1 Case 2.3.1 2 3 1 Case 2.3.2a 2 3 1 Case 2.3.2b And symmetrical cases bending towards 5-6-7-8 that will cover a heavy emerald among 5-6-7-8 or a light one among 1-2-3-4 1 2 is too heavy1 is too heavy 3 is too heavy Weighting 1Weighting 2Weighting 3

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