Henry Ernest Dudeney/Puzzles and Curious Problems/285 - The Teashop Check/Solution

by : $285$

 * The Teashop Check

Solution
One of the customers' bills was $7 \oldpence$, and the other's was $3 \shillings 4 \tfrac 1 2 \oldpence$

Proof
First note that the total of all the prices is $3 \shillings 11 \tfrac 1 2 \oldpence$.

This is what you would get if all the holes were punched.

$7 \oldpence$ can be punched as follows:

For every one of these punch patterns, there is a directly complementary one which selects all the opposite cash values.

These all add up to $3 \shillings 11 \tfrac 1 2 \oldpence - 7 \oldpence$, or $3 \shillings 4 \tfrac 1 2 \oldpence$

Hence the result.