Henry Ernest Dudeney/Modern Puzzles/1 - Concerning a Cheque/Solution

by : $1$

 * Concerning a Cheque

Solution
As puts it:


 * If you set to work under the notion that there were only pounds and shillings -- no pence --- in the amount, a solution is impossible.

The amount on the cheque was $\pounds 5 \ 11 \shillings 6 \oldpence$

He received $\pounds 11 \ 5 \shillings 6 \oldpence$

After spending half a crown, which is $2 \shillings 6 \oldpence$, he had $\pounds 11 \ 3 \shillings$

This is twice as much as $\pounds 5 \ 11 \shillings 6 \oldpence$

Proof
We recall the conversion factors:


 * $12$ pennies make one shilling


 * $20$ shillings make one pound.


 * $1$ half crown consists of $2$ shillings and $6$ pence.

Let $C$ be the amount written on the cheque.

Let $C'$ be the amount paid out to the man by the bank.

Let $P$ be the final amount in the man's pocket.

Suppose $C$ were an integral number of shillings.

Then $C'$ would also be an integral number of shillings.

But $P$, which is $C'$ less $2 \tfrac 1 2$ shillings is then not an integral number of shillings.

That means $P$ could not be twice $C$.

Hence 's warning.

Let $C$ consist of $C_l$ pounds, $C_s$ shillings and $C_d$ pennies.

Let $P$ consist of $P_l$ pounds, $P_s$ shillings and $P_d$ pennies.

But what we can say about the pennies is that:
 * $(1): \quad 2 \times \paren {C_d - 6} = P_d \pmod {12}$

because $2 \paren {C - 2 \shillings 6 \oldpence} = P$

and also that:
 * $(2): \quad C_d - 6 = P_d \pmod {12}$

because $P_d$ is the actual number of pennies.

from which it follows that $C_d = 6$.

Let $D_1$ be the value of the cheque in pennies.

Let $D_2$ be the money the man left the bank with in pennies.

Let $D_3$ be the money the man arrived home with in pennies.

We have:
 * $D_1 = 240 C_l + 12 C_s + 6$

After coming out of the bank, the man has:
 * $D_2 = 240 C_s + 12 C_l + 6$

After arriving home, the man has:
 * $D_3 = 240 C_s + 12 \paren {C_l - 2}$

But we have:
 * $D_1 \times 2 = D_3$

which leads us to:

By inspection, we arrive at:
 * $13 \times 5 = 65 = 6 \times 11 - 1$

and the answer follows.