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

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Puzzles and Curious Problems by Henry Ernest Dudeney: $285$

The Teashop Check
We give an example of the check supposed to be used at certain popular teashops.
The waitress punches holes in the tickets to indicate the amount of the purchase.
$\boxed {\begin{array} {rcl} \\

\tfrac 1 2 \oldpence & --- & \bullet \\ 1 \oldpence & --- \\ 1 \tfrac 1 2 \oldpence & --- \\ 2 \oldpence & --- \\ 2 \tfrac 1 2 \oldpence & --- \\ 3 \oldpence & --- & \bullet \\ 4 \oldpence & --- \\ 6 \oldpence & --- \\ 7 \oldpence & --- \\ 8 \oldpence & --- \\ 1 \shillings & --- \\ 

& & \\

\end{array} }$

Thus, in the example, the two holes indicate that the customer has to pay $3 \tfrac 1 2 \oldpence$
But the girl might, if she had chosen, have punched in any one of three other ways --
$2 \tfrac 1 2 \oldpence$ and $1 \oldpence$, or $2 \oldpence$ and $1 \tfrac 1 2 \oldpence$, or $2 \oldpence$, $1 \oldpence$ and $\tfrac 1 2 \oldpence$
On one occasion a waitress said, "I can punch this ticket in any one of $10$ different ways, and no more."
Her coworker, whose customer owed a different amount, said, "Same here."
What were the amounts of the purchases of each of their customers?
Only one hole is allowed to be punched against any given amount.


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:

\(\text {(1)}: \quad\) \(\ds \) \(\) \(\ds 7 \oldpence\)
\(\text {(2)}: \quad\) \(\ds \) \(\) \(\ds {6 \oldpence} + {1 \oldpence}\)
\(\text {(3)}: \quad\) \(\ds \) \(\) \(\ds {4 \oldpence} + {3 \oldpence}\)
\(\text {(4)}: \quad\) \(\ds \) \(\) \(\ds {4 \oldpence} + {2 \tfrac 1 2 \oldpence} + {\tfrac 1 2 \oldpence}\)
\(\text {(5)}: \quad\) \(\ds \) \(\) \(\ds {4 \oldpence} + {2 \oldpence} + {1 \oldpence}\)
\(\text {(6)}: \quad\) \(\ds \) \(\) \(\ds {4 \oldpence} + {1 \tfrac 1 2 \oldpence} + {1 \oldpence} + {\tfrac 1 2 \oldpence}\)
\(\text {(7)}: \quad\) \(\ds \) \(\) \(\ds {3 \oldpence} + {2 \tfrac 1 2 \oldpence} + {1 \tfrac 1 2 \oldpence}\)
\(\text {(8)}: \quad\) \(\ds \) \(\) \(\ds {3 \oldpence} + {2 \tfrac 1 2 \oldpence} + {1 \oldpence} + {\tfrac 1 2 \oldpence}\)
\(\text {(9)}: \quad\) \(\ds \) \(\) \(\ds {3 \oldpence} + {2 \oldpence} + {1 \tfrac 1 2 \oldpence} + {\tfrac 1 2 \oldpence}\)
\(\text {(10)}: \quad\) \(\ds \) \(\) \(\ds {2 \tfrac 1 2 \oldpence} + {2 \oldpence} + {1 \tfrac 1 2 \oldpence} + {1 \oldpence}\)


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.

$\blacksquare$


Sources

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