Henry Ernest Dudeney/Puzzles and Curious Problems/302 - Delivering the Milk/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $302$

A milkman one morning was driving to his dairy with two $10$-gallon cans full of milk,

when he was stopped by two countrywomen, who implored him to sell them a quart of milk each.

Mrs. Green had a jug holding exactly $5$ pints, and Mrs. Brown a jug holding exactly $4$ pints,

but the milkman had no measure whatsoever.

How did he manage to put an exact quart into each of the jugs?

It was the second quart that gave all the difficulty.

But he contrived to do it in as few as nine transactions --

and by a "transaction" we mean the pouring from a can into a jug, or from one jug to another, or from a jug back to the can.

How did he do it?

To simplify understanding, let us convert all measurements to pints.

Let $C_1$ and $C_2$ denote the two $10$-gallon cans, each of which hold $80$ pints.

Let $J_5$ and $J_4$ denote the $5$ pint and $4$ pint jugs repectively.

Proceed as follows:

$(1): \quad$ Fill $J_5$ from $C_1$.

$(2): \quad$ Fill $J_4$ from $J_5$.

$(3): \quad$ Empty $J_4$ back into $C_1$.

$(4): \quad$ Empty $J_5$ into $J_4$.

$(5): \quad$ Fill $J_5$ from $C_1$.

$(6): \quad$ Fill $J_4$ from $J_5$.

$(7): \quad$ Empty $J_4$ back into $C_1$.

At this stage, $J_5$ contains the first required quart, $C_2$ is still full, $J_4$ is empty, and $C_1$ is $1$ quart short of full.

$(8): \quad$ Fill $J_4$ from $C_2$.

$(9): \quad$ Fill $C_1$ from $J_4$.

As $C_1$ was $1$ quart, that is $2$ pints, short of full, $J_4$ now holds the second required $2$ quarts.