Henry Ernest Dudeney/Modern Puzzles/183 - At the Brook/General Solution

by : $183$

 * At the Brook

General Solution
starts by demonstrating that any integer quantity from $1$ to $16$ pints can be measured using these two containers.

Perform the transactions according to the one of the following two procedures:


 * $\begin {array} {r|c|c|c} & \text {$15$-pint} & \text {$16$-pint} \\ \hline

& 0 & 0 \\ & 0 & 1 \\ & 1 & 0 \\ & 1 & 16 \\ & 0 & 2 \\ & 2 & 0 \\ & 2 & 16 \\ & 0 & 3 \\ & 3 & 0 \\ & 3 & 16 \\ & 0 & 4 \\ & 4 & 0 \\ & 4 & 16 \\ & 0 & 5 \\ & 5 & 0 \\ & 5 & 16 \\ & 0 & 6 \\ & 6 & 0 \\ & 6 & 16 \\ & 0 & 7 \\ & 7 & 0 \\ \end {array} \qquad \begin {array} {r|c|c|c} & \text {$15$-pint} & \text {$16$-pint} \\ \hline & 0 & 0 \\ & 0 & 15 \\ & 15 & 15 \\ & 14 & 0 \\ & 0 & 14 \\ & 15 & 14 \\ & 13 & 0 \\ & 0 & 13 \\ & 15 & 13 \\ & 12 & 0 \\ & 0 & 12 \\ & 15 & 12 \\ & 11 & 0 \\ & 0 & 11 \\ & 15 & 11 \\ & 10 & 0 \\ & 0 & 10 \\ & 15 & 10 \\ & 9 & 0 \\ & 0 & 9 \\ & 15 & 9 \\ \end {array}$
 * & 0 & 16 \\
 * & 15 & 1 \\
 * & 15 & 2 \\
 * & 15 & 3 \\
 * & 15 & 4 \\
 * & 15 & 5 \\
 * & 15 & 6 \\
 * & 15 & 7 \\
 * & 15 & 0 \\
 * & 14 & 16 \\
 * & 13 & 16 \\
 * & 12 & 16 \\
 * & 11 & 16 \\
 * & 10 & 16 \\
 * & 9 & 16 \\
 * & 8 & 16 \\

In the first column, the asterisks mark where the quantities from $1$ pint to $8$ pints have been measured, and a continuance of the pattern for another few transactions.

In the second column, the asterisks mark where the quantities from $15$ pints down to $8$ pints have been measured.

It is seen that fewer transactions are needed to get to $8$ by filling the $15$-pint container first, while to get to $7$ it is quicker to start by filling the $16$-pint container.