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

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Modern Puzzles by Henry Ernest Dudeney: $183$

At the Brook
A man goes to the brook with two measures of $15$ pints and $16$ pints.
How is he to measure exactly $8$ pints of water, in the fewest possible transactions?
Filling or emptying a vessel or pouring any quantity from one vessel to another counts as a transaction.


Solution

Perform the transactions according to the following:

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

& 0 & 0 \\ (1): & 15 & 0 \\ (2): & 0 & 15 \\ (3): & 15 & 15 \\ (4): & 14 & 16 \\ (5): & 14 & 0 \\ (6): & 0 & 14 \\ (7): & 15 & 14 \\ (8): & 13 & 16 \\ (9): & 13 & 0 \\ (10): & 0 & 13 \\ (11): & 15 & 13 \\ (12): & 12 & 16 \\ (13): & 12 & 0 \\ (14): & 0 & 12 \\ (15): & 15 & 12 \\ (16): & 11 & 16 \\ (17): & 11 & 0 \\ (18): & 0 & 11 \\ (19): & 15 & 11 \\ (20): & 10 & 16 \\ (21): & 10 & 0 \\ (22): & 0 & 10 \\ (23): & 15 & 10 \\ (24): & 9 & 16 \\ (25): & 9 & 0 \\ (26): & 0 & 9 \\ (27): & 15 & 9 \\ (28): & 8 & 16 \\ \end {array}$

and it is seen that $8$ pints have been measured out in $28$ transactions.


General Solution

Dudeney 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 & 16 \\
  • & 15 & 1 \\
& 0 & 1 \\
& 1 & 0 \\
& 1 & 16 \\
  • & 15 & 2 \\
& 0 & 2 \\
& 2 & 0 \\
& 2 & 16 \\
  • & 15 & 3 \\
& 0 & 3 \\
& 3 & 0 \\
& 3 & 16 \\
  • & 15 & 4 \\
& 0 & 4 \\
& 4 & 0 \\
& 4 & 16 \\
  • & 15 & 5 \\
& 0 & 5 \\
& 5 & 0 \\
& 5 & 16 \\
  • & 15 & 6 \\
& 0 & 6 \\
& 6 & 0 \\
& 6 & 16 \\
  • & 15 & 7 \\
& 0 & 7 \\
& 7 & 0 \\

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

& 0 & 0 \\
  • & 15 & 0 \\
& 0 & 15 \\
& 15 & 15 \\
  • & 14 & 16 \\
& 14 & 0 \\
& 0 & 14 \\
& 15 & 14 \\
  • & 13 & 16 \\
& 13 & 0 \\
& 0 & 13 \\
& 15 & 13 \\
  • & 12 & 16 \\
& 12 & 0 \\
& 0 & 12 \\
& 15 & 12 \\
  • & 11 & 16 \\
& 11 & 0 \\
& 0 & 11 \\
& 15 & 11 \\
  • & 10 & 16 \\
& 10 & 0 \\
& 0 & 10 \\
& 15 & 10 \\
  • & 9 & 16 \\
& 9 & 0 \\
& 0 & 9 \\
& 15 & 9 \\
  • & 8 & 16 \\

\end {array}$

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.


Historical Note

Martin Gardner reports on his Mathematical Games article in the September $1963$ issue of Scientific American, where he describes a technique for solving liquid pouring problems using isometric paper.

The subject is also discussed in 1965: T.H. O'Beirne: Puzzles and Paradoxes.


Sources

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