One Light Column

There are ten columns of pennies, ten coins to a column. Every coin is genuine and weighs $10$ grams, except that one whole column is counterfeit, each of its coins weighing $9$ grams. You have an accurate scale but may use it for only a single weighing. How do you find the counterfeit column, and with how few coins?

Solution

Number the columns $0$ to $9$, and take that many coins from each: none from column $0$, one from column $1$, two from column $2$, and so on up to nine from column $9$. That is $0 + 1 + 2 + \dots + 9 = 45$ coins on the scale at once.

Were every coin genuine, the pile would weigh $450$ grams. Each counterfeit coin is a gram light, and the number of counterfeit coins in the pile is exactly the number of the offending column. So the shortfall below $450$ grams, read straight off the scale, names the column: a deficit of $7$ grams means column $7$, and a deficit of nothing at all means column $0$, from which we took no coin. One weighing settles it, using $45$ coins. (Taking $1$ through $10$ coins instead also works, but costs ten more coins for no extra information.)