Henry Ernest Dudeney/Puzzles and Curious Problems/98 - Digital Money/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $98$

Digital Money

Every letter in the following multiplication represents one of the digits, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, all different.

What is the value obtained if $K = 8$?

  A   B   C
x         K
-----------
 DE  FG   H

Solution

   3   7   5
x          8
------------
£ 26  19   4

Proof

Given that $K = 8$, recalling that $1$ shilling is $12$ pence:

The possible values of $H$ are $0, 8$ or $4$.

Since both $0, 8$ are not available, we have $H = 4$.

This is only possible if $C = 2, 5$ or $8$.

Again, $8$ is not available, so $C = 2$ or $5$.

This provides a carry of $1$ or $3$, respectively, to the shillings place.

Without the carry, with $K = 8$, recalling that $1$ pound is $20$ shillings, the possible values of $FG$ are:

$8, 16, 4, 12, 0$

With the carries, the possible values of $FG$ are:

$9, 17, 5, 13, 1, 11, 19, 7, 15, 3$

Since $FG$ is a $2$-digit number, the only possible values of $FG$ are:

$17, 13, 19, 15$

which results from:

$\set {12, 16} + \set {1, 3}$

so the possible values of $B$ are:

$2, 4, 7, 9$

Since $4$ is taken, we check all cases where:

$C = 2$ and $B = 7$ or $9$

$C = 5$ and $B = 2, 7, 9$

This needs considerable tedious hard slog to complete it. In particular: check that $(B,C) = (7,2)$ and $(9,5)$ are not possible; loop all values of $A$ and see whether they result in repeated digits. The solution is unique. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $98$. -- Digital Money