Solution to February’s Puzzle: Counterfeit Coins

This puzzle is from Clifford Jensen. Suppose you have 10 stacks of coins, with 10 coins in each stack. One stack consists of 10 counterfeit coins and the other 9 stacks each consist of 10 legitimate coins. Each legitimate coin weighs exactly 1 gram. Each counterfeit coin weighs exactly 0.9 grams. You have a digital scale that’s graduated in tenths of grams. Using the scale to take only one reading, determine which stack has the 10 counterfeit coins. You can weigh any number of coins from any number of stacks, but must you weigh them all together (i.e., you can take only one reading from the scale).

Take 1 coin from stack #1, 2 coins from stack #2, and so on. Weigh the stack of 55 coins. If all the coins were legitimate, the scale would show 55 grams. If stack #3 is the stack of counterfeit coins, the scale will show 54.7 grams because the pile of coins you weighed contains 3 counterfeit coins and is therefore 0.3 grams light. More generally, if stack n is the stack of counterfeit coins and w is the weight the scale shows, then n = (55.0 - w)/0.1.

March’s Puzzle: Too Clever by Half

A chicken and a half lay an egg and a half in a day and a half. How many eggs would one chicken lay in three days? A builder and a half build a house and a half in a year and a half using a tool and a half. How many houses would one builder build in nine years? Can you generalize your calculation to solve both equations?