Henry Ernest Dudeney/Puzzles and Curious Problems/296 - Water Measurement/Solution

by : $296$

 * Water Measurement
 * A maid was sent to the brook with two vessels that exactly measured $7$ pints and $11$ pints exactly.
 * She had to bring back exactly $2$ pints of water.
 * What is the smallest possible number of transactions necessary?


 * A "transaction" is filling a vessel, of emptying it, or pouring from one vessel to another.

Solution

 * $14$ transactions are needed.

Proof
Let $V_7$ and $V_{11}$ denote the $7$-pint and $11$-pint vessels respectively.

In the below, the successive lines indicate the status of each of $V_7$ and $V_{11}$:


 * $\begin{array}{c|c} V_7 & V_{11} \\ \hline

7 & 0 \\ 0 & 7 \\ 7 & 7 \\ 3 & 11 \\ 3 & 0 \\ 0 & 3 \\ 7 & 3 \\ 0 & 10 \\ 7 & 10 \\ 6 & 11 \\ 6 & 0 \\ 0 & 6 \\ 7 & 6 \\ 2 & 11 \\ \end{array}$

and it is seen that the required $2$ pints are in the $7$-pint vessel.