The following is a famous problem of Bachet as recounted by Heinrich Dörrie in his book *100 Great Problems of Elementary Mathematics*:

“A merchant had a forty-pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used [in a balance scale] to weigh every integral weight between 1 and 40 pounds [when we are allowed to put a weight in either of the two pans]. What were the weights of the pieces?

(This problem stems from the French mathematician Claude Gaspard Bachet de Méziriac (1581-1638), who solved it in his famous book *Problèmes plaisants et délectables qui se font par les nombres*, published in 1624.)”

The problem has a nice solution using ternary numbers. See the Weight Problem of Bachet.

**(Update 4/10/2019)**I made a mistake in my original write-up, which I realized after discovering that Leonardo of Pisa (aka Fibonacci) had posed and solved this problem 400 years before Bachet in his 1202 AD book *Liber Abaci.* The PDF file has been updated. (Why didn’t anyone catch that and tell me? You are being too polite. You have to leave your ego at the door if you are going to be involved with mathematics.)