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The Eleven Gemstones

(This is a very easy problem, IMO, but it's fun, at least I think so)

The Mine Overlord passes away, leaving his 11 precious gemstones that he has mined over the years to the Chief-Miner, the Sub-Chief-Miner and the Minor-Miner-Squadron-Leader.

He declares that the Chief-Miner, being the highest ranked, shall receive half of the gemstones. The Sub-Chief-Miner, the next in rank, shall receive a third of the gemstones, and the Minor-Miner-Squadron-Leader shall receive a twelfth. Anything left over, if there is anything, goes to charity.

How can they divide the eleven precious gemstones into halves, thirds and twelfths??? Frustrated, they call the famous Jewel Mathematician, who comes with her priceless gemstone brooch. "Easy," she says, adding her own gemstone to the pile. "There are twelve. Chief-Miner, you take 6, which is half. Sub-Chief-Miner, you take 4, which is a third. Minor-Miner-Squadron-Leader, you take 1, which is a twelfth. That's the 11 gemstones, and I'll take my own gemstone brooch back."

She then departs, leaving them confused, yet divided up nicely, with her still being able to keep her gemstone brooch.

***

What is the hole here?

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The Eleven Gemstones

(This is a very easy problem, IMO, but it's fun, at least I think so)

The Mine Overlord passes away, leaving his 11 precious gemstones that he has mined over the years to the Chief-Miner, the Sub-Chief-Miner and the Minor-Miner-Squadron-Leader.

He declares that the Chief-Miner, being the highest ranked, shall receive half of the gemstones. The Sub-Chief-Miner, the next in rank, shall receive a third of the gemstones, and the Minor-Miner-Squadron-Leader shall receive a twelfth. Anything left over, if there is anything, goes to charity.

How can they divide the eleven precious gemstones into halves, thirds and twelfths??? Frustrated, they call the famous Jewel Mathematician, who comes with her priceless gemstone brooch. "Easy," she says, adding her own gemstone to the pile. "There are twelve. Chief-Miner, you take 6, which is half. Sub-Chief-Miner, you take 4, which is a third. Minor-Miner-Squadron-Leader, you take 1, which is a twelfth. That's the 11 gemstones, and I'll take my own gemstone brooch back."

She then departs, leaving them confused, yet divided up nicely, with her still being able to keep her gemstone brooch.

***

What is the hole here?

1/2 + 1/3 + 1/12 = 11/12.

Adding a stone to get 12 permits the 11 to be divided in the right proportion and leaves the 12th stone undistributed.

This also works with 17 using 1/2, 1/3 and 1/9 [17/18]

and with 19 using 1/2, 1/4 and 1/5 [19/20].

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