Four Weights for Forty

A stone weighing forty pounds is broken into four pieces. A shopkeeper finds that with these four pieces, placed in either pan of a balance, he can weigh out any whole number of pounds from $1$ to $40$. What are the weights of the four pieces?

Solution

The four pieces weigh $1$, $3$, $9$ and $27$ pounds, the powers of three, and indeed $1 + 3 + 9 + 27 = 40$.

The reason they suffice is that a balance lets each weight do one of three things: sit in the pan opposite the load, sit in the same pan as the load, or stay off the scale altogether. So the amounts you can weigh are exactly the totals $$(\pm 1) + (\pm 3) + (\pm 9) + (\pm 27),$$ where each of the four pieces may be added (placed opposite the load), subtracted (placed beside it), or left out. This is what is called balanced ternary, counting in threes with the digits $-1$, $0$ and $1$. Every whole number from $-40$ to $40$ can be written in this form in exactly one way, so in particular each amount from $1$ to $40$ can be weighed, and uniquely. For example $5 = 9 - 3 - 1$: put the $9$ opposite the load, the $3$ and the $1$ beside it.