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.

Divide the 200 coins into 3 groups of 66 (666666), 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.

Divide the group of 66 into 3 groups of 22 (222222), and do the same thing.

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 (888).

Weigh 3 against 3 (332).

Weigh 1 against 1 (this works either with 2 or 3 coins).

Group the 200 into 676766. Weigh 67 against 67, which will result either in a group of 66 or 67 that has the Canadian coin.

Group as 222223 or 222222. Either way, weigh 22 against 22, which results in either a group of 22 or a group of 23 that has the Canadian coin.

If 22, split 778; for 23, split 887. Weigh 7 against 7 or 8 against 8, resulting in a group or 7 or 8 with the Canadian coin

If 7, split 223; if 8, split 332; weigh 2 against 2 or 3 against 3. You will now have a group of either 2 or 3 with the Canadian coin.

If 3, split 111, if 2 split 11.
Solution 3: Michael K.

Weigh 81 coins against 81 coins (leaving 38 coins  818138). 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.

Weigh 27 against 27 to find a group of 27 with the Canadian coin (272727).

Weigh 9 against 9 (999).

Weigh 3 against 3 (333).

Weigh 1 against 1 (111).
Posted November 06, 2018