Inside SFS

November 2018 Math Challenge Solutions

Original Puzzle:

Weighing Different Coins
Milo has 200 quarters, and only 1 of them is a Canadian quarter, which is lighter than the American quarter. “I want to find out which is my Canadian quarter without having to look at one quarter at a time! I have a balance with two pans (see photo) and want to use that!” Can you help him find the Canadian quarter with no more than 5 different times weighing the coins? How?
You might want to consider this easier version of the same problem, and think of a strategy:
 
Suppose he had only 7 coins, where 6 are the same weight and 1 is lighter than the others. Using a balance with two pans (see photo), what is the least number of times you need to weigh the coins to determine which coin is lighter?
Solution 1: Anders K.
  1. Divide the 200 coins into 3 groups of 66 (66-66-66), and put the 2 extra coins to the side. Weigh 2 groups of 66 against each other. If one of the sides is lighter, that side has the Canadian coin; otherwise, assume the unweighed group has the Canadian coin.
  2. Divide the group of 66 into 3 groups of 22 (22-22-22), and do the same thing.
  3. Add the extra 2 coins from step 1 to the suspect group of 22. Now you have a group of 24. Weigh 8 against 8 to isolate a group of 8 (8-8-8).
  4. Weigh 3 against 3 (3-3-2).
  5. Weigh 1 against 1 (this works either with 2 or 3 coins).
Solution 2: Soren K./Alexx C.
  1. Group the 200 into 67-67-66. Weigh 67 against 67, which will result either in a group of 66 or 67 that has the Canadian coin.
  2. Group as 22-22-23 or 22-22-22. Either way, weigh 22 against 22, which results in either a group of 22 or a group of 23 that has the Canadian coin.
  3. If 22, split 7-7-8; for 23, split 8-8-7. Weigh 7 against 7 or 8 against 8, resulting in a group or 7 or 8 with the Canadian coin
  4. If 7, split 2-2-3; if 8, split 3-3-2; weigh 2 against 2 or 3 against 3. You will now have a group of either 2 or 3 with the Canadian coin.
  5. If 3, split 1-1-1, if 2 split 1-1.
Solution 3: Michael K.
  1. Weigh 81 coins against 81 coins (leaving 38 coins - 81-81-38). If the Canadian coin is in the group of 38, add 43 coins to it from one of the other groups of all American coins. Regardless, you will end up with a group of 81 coins with the Canadian coin.
  2. Weigh 27 against 27 to find a group of 27 with the Canadian coin (27-27-27).
  3. Weigh 9 against 9 (9-9-9).
  4. Weigh 3 against 3 (3-3-3).
  5. Weigh 1 against 1 (1-1-1).

Posted November 06, 2018

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