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You have 11 visually identical coins. One of them is marked genuine (and you know which one this is). Out of the remaining 10 coins, all but one of them are also genuine, but you don't know which ones. The remaining odd coin has a different weight, either heavier or lighter than the others.

You have a regular two-pan balance. This works like you expect: you load a certain number of coins to each side, and the balance will tell you the heavier pan (or whether they have equal weight).

How many weighings are necessary in the worst case to find the odd coin and determine whether it weighs more or less than genuine coins?

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  • $\begingroup$ If they're all visually identical, how is one of them "marked genuine"? $\endgroup$ – dpwilson Jan 15 '16 at 21:00
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    $\begingroup$ I'm confused, the title says "Find the other genuine coin", but we're looking for the one fake coin, not the one genuine coin. $\endgroup$ – Ninety-Three Jan 15 '16 at 21:02
  • $\begingroup$ @bleh you say 'find the other genuine coin' but say later 'determine whether it weighs more or less than genuine coins'. $\endgroup$ – Beastly Gerbil Jan 15 '16 at 21:18
  • $\begingroup$ Corrected the title to the most reasonable interpretation, since the puzzle itself makes sense. $\endgroup$ – Zerris Jan 15 '16 at 22:10
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We can determine in three weighings every time.

Method:

Label the coins 1 - 11, with 1 being genuine

Weighing 1: 1, 2, 3, 4 vs. 5, 6, 7, 8

If the weights are the same, then

Weighing 2a: 9 vs. 10

If they're the same again, then

Weighing 3a: 1 vs. 11 - we know 11 is fake, but this tells us if it's heavier or lighter

If the left side was heaver in Weighing 2a, then

Weighing 3b: 1 vs. 9 - if they're the same, then 10 is fake and light, otherwise 9 is fake and heavy

If the right side was heaver in Weighing 2a, then

Weighing 3c: 1 vs. 9 - if they're the same, then 10 is fake and heavy, otherwise 9 is fake and light

If the left side was heavier in Weighing 1:

Weighing 2b: 5, 6 vs. 7, 8

If they're the same, then

Weighing 3d: 2 vs. 3 - whichever is heavier is also fake, if they're the same then 4 is heavy and fake

If the left side was heavier in Weighing 2b, then

Weighing 3e: 7 vs 8 - whichever is lighter is also fake

If the right side was heavier in Weighing 2b, then

Weighing 3f: 5 vs 6 - whichever is lighter is also fake

If the right side was heavier in Weighing 1:

Weighing 2c: 5, 6 vs. 7, 8

If they're the same, then

Weighing 3g: 2 vs. 3 - whichever is lighter is also fake, if they're the same then 4 is light and fake

If the left side was heavier in Weighing 2c, then

Weighing 3h: 5 vs 6 - whichever is heavier is also fake

If the right side was heavier in Weighing 2c, then

Weighing 3i: 7 vs 8 - whichever is heavier is also fake

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