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well, sapphires must be the minimum, so set that to 1.

to have more than twice that for emeralds, emeralds must be > 2, so set that to 3.

To have more than twice that for diamonds, diamonds must be >6, so set that to 7.

1+3+7 = 11 total.

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1 sapphire < 1/2 emeralds

1 diamond > 2 emeralds

I'm assuming you can have 0 of a gem, so in that case you would have:

0 sapphires, 1 emerald, and 3 diamonds = 4 gems

If there its garunteed that the king has least of one each gem when its said that he has "a number" of each gem, then:

1 sapphire, 3 emeralds, 7 diamonds = 11 gems

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the answer is 14.... the number of saphires is the least, and theres at least 1. therefore the number of emeralds has to be at least 4, since half of 4 is 2 and less than that is 1. Since there are 4 emeralds, there must be at least 9 diamonds, since it says there are more than 2x the diamonds as emeralds. 9 + 4 + 1 = 14

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