Definition:Donkey and Mule Problem

Definition

A Donkey and Mule problem is of the form:

$2$ participants are comparing their quantity of goods they own relative to the other.

Participant $1$ suggests transferring some of the goods of one of them to the other, and remarks on their relative quantities after that exchange.

Participant $2$ then suggests the same (or a similar) thing, but transferring the goods in the other direction, and again remarks on their relative quantities after that exchange.

From these remarks, $2$ simultaneous linear equations can be set up and (assuming the question is well-crafted) solved.

The name comes from the original setting of this problem concerning a donkey and a mule each carrying a number of sacks.

Also known as

A Donkey and Mule problem is usually referred to as an Ass and Mule problem, but unfortunately the meaning of the word ass has been so corrupted in modern times that to many people it is an obscenity referring to an unmentionable body part.

Also see

• Results about Donkey and Mule problems can be found here.

Historical Note

The earliest known occurrence of a Donkey and Mule problem is in a piece of Greek verse attributed to Euclid.

Since then it has appeared in many places, and crops up regularly in puzzle anthologies.

The general form:

Let $\tuple {a, b, c, d}$ denote the general problem:

If I had $a$ from you I'd have $b$ times as many as you.

If I had $c$ from you I'd have $d$ times as much as you.

appears never to have been completely explored.