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Coins #1
       The way to approach this type of problem is to start with just a single coin, and work up towards the maximum.
       With just one coin, there is nothing you can do with just a pan balance. However, in part (B) you know that single coin is the false one, and in part (C) you can compare it to a real coin and determine if it is lighter or heavier in 1 weighing.
       With 2 coins you can compare them. If they balance, you know both are genuine. If not, you cannot tell which is the fake. In part (C) you can compare either coin to a real one. If they balance, then the other coin is the fake, otherwise the first coin is the fake.
       With 3 coins you can finally be sure. You compare 2 coins. If they balance, then they are both genuine, and the third coin is potentially fake. You check that by weighing it against one of the first 2 coins. This takes 2 weighings. In parts (B) and (C) you can skip the second weighing. If the 2 coins do not balance, then the third coin must be genuine, so you check either of the first two against the third.
       With 4 coins you finally need to make a decision. If you weigh 1 coin against 1, then if they balance you are left with 2 coins, and it will take 2 more weighings to determine the false coin. If you weigh 2 against 2 and they don't balance, then you need 2 more weighings to determine the false coin. This does not happen in case (C) because you can weigh 2 unknown coins against 1 unknown coin plus one genuine coin. If they don't match, then it will only take 1 more weighing to determine which of the 3 is the false coin. Or, you could weigh 3 of the unknown coins against 3 genuine coins.
       With 5 coins you compare 2 against 2. If they balance, you need 1 weighing for the last coin. If they do not balance, you need 2 more weighings. In case (C) you still need only 2 weighings. You start with 2 unknowns against 1 unknown and one genuine coin. If they don't match, you weigh the first 2 against each other. If they do match, it only takes 1 weighing for the remaining 2 coins.
       6 coins is the same as 5, except that in case (C) you will now need 3 weighings. For 7 and 8 coins you can still start with 2 against 2, and determine the fake among the last 3 or 4 in 2 weighings.
       With 9 or 10 coins you start with 3 against 3. If they do not balance, then you have 3 that might be light, and 3 that might be heavy. You next balance 2 light and 1 heavy against 2 light and 1 heavy. If they balance, then you only have 2 coins left. If one side is heavier, then either the false coin is the heavy one in that pan, or one of the light ones in the other pan, so you balance those 2 lights to find out. In all cases it takes no more than 3 weighings altogether.
       For 11 or 12 coins you start by weighing 4 against 4. If they balance, you only have 3 or 4 left, which takes 2 weighings. If not, then you have 4 potential light coins, and 4 potential heavy coins. You proceed exactly as for 10 coins.
       For 13 coins you can identify the false coin if you know that there is one, but you cannot always tell if it is heavy or light. For 14 coins in case (C) you can begin by weighing 5 unknown coins against 4 unknowns plus one genuine coin. If they do not balance, then you have 5 possible light coins and 4 possible heavy coins, or vice-versa. Either way, you proceed as for 10 coins.
       Final answer, (A) 12 coins, (B) 13 coins, (C) 14 coins.

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