Answers to selected weights and measures puzzles

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Scale Riddle Answers The Contest Center 59 DeGarmo Hills Road Wappingers Falls, NY 12590

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Pan Balance
       Many people immediately jump to the answer that you should buy weights of 1, 2, 4, 8, 16, etc. ounces. This is because they learned in school that these can be added to make any whole number, for example 21 is 16+4+1. So you could weigh a 21-ounce object by putting it in one pan, and placing weights 1, 4 and 16 in the other pan.
       People who consider the problem more carefully will realize that you can put the weights on either pan. Using weights 1, 3, 9, 27, 81, etc. you can add and subtract them to make any whole number, for example 21 is 27+3-9. So you could weigh a 21-ounce object by placing it in one pan along with the 9-ounce weight, and putting 3 and 27 in the other pan.

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 the 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.

Coins #2
       Take 1 coin from the first bag, 2 coins from the second bag, etc. through 20 coins from the twentieth bag. This should total 2100 grams. The number of grams short of 2100 tells you the bad bag.

U.S. Coins
       The denominations should be 1 cent, 3 cents, 10 cents, 30 cents, 1 dollar, 3 dollars, 10 dollars, 30 dollars and 100 dollars.

Cups
       Here is one way: fill the 9-gill cup, and pour it off into the 4-gill cup until it is full. This leaves 5 gills. Empty the 4-gill cup and fill it again from the 9-gill cup. This leaves 1 gill in the 9-gill cup. Empty the 4-gill cup and pour in the 1 gill from the other cup. Now refill the 9-gill cup and pour it into the 4-gill cup until it is full. This leaves 6 gills in the 9-gill cup.
       Here is a different way: fill the 4-gill cup and pour it into the 9-gill cup. Repeat, so there are 8 gills in the 9-gill cup. Fill the 4-gill cup again and pour it into the 9-gill cup, filling the 9-gill cup and leaving 3 gills in the 4-gill cup. Empty the 9-gill cup and pour in the 3 gills. Now fill the 4-gill cup 2 more times and pour it into the 9-gill cup until it is full. This will leave 2 gills in the 4-gill cup. Empty the 9-gill cup, pour in the 2 gills, and then 4 more gills from the 4-gill cup.

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