1250.11 – False Coin no. 1

You are applying for a job as inspector at the U.S. Mint. You have been handed 8 gold coins and are told that one of them is counterfeit: it is slightly lighter than each of the other seven. You are given a two-pan balance scale, and you are told that you may weigh coins on that scale twice. How can you identify the counterfeit coin?

Your prospective employers are fussy about the scale; if you put some coins on for the first weighing, and then change the coins in any way, that counts right there as your second weighing.

Solution

Call the coins (in three artificial groups) $A$ , $B$ , and $C$ ; $a$ , $b$ , and $c$ ; and finally $X$ and $Y$ .

1st weighing: $A, B, C$ against $a, b, c$ .

Case 1: $A, B, C$ is heavier. Then the light coin is $a$ , $b$ , or $c$ .
- 2nd weighing: $a$ against $b$ . If $a = b$ , $c$ is the light one.
Case 2: $a, b, c$ is heavier. Then the light coin is $A$ , $B$ , or $C$ .
- Proceed as in Case 1, but with $A, B, C$ instead of $a, b, c$ .
Case 3: $A, B, C$ = $a, b, c$ . Then $X$ or $Y$ is the light one.
- 2nd weighing: X against Y.