Henry Ernest Dudeney/Puzzles and Curious Problems/285 - The Teashop Check/Solution
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
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $285$. -- The Teashop Check