Henry Ernest Dudeney/Modern Puzzles/52 - The Five Cards/Solution

by : $52$

 * The Five Cards

Solution

 * $\boxed 3 \boxed 9 \ \boxed 1 \ \boxed 5 \boxed 7$

or:
 * $\boxed 5 \boxed 7 \ \boxed 1 \ \boxed 3 \boxed 9$

Proof
Let $d_1$ and $d_2$ be the two $2$-digit numbers at either end.

Let $s$ be the single-digit subtrahend.

Let $n$ be the repdigit that results from $d_1 \times d_2 - s$.

Suppose $n$ is odd.

Then $n + s = d_1 \times d_2$ is even.

But because $d_1$ and $d_2$ are both odd, this is impossible.

So $n$ is even.

At this stage we do not know whether $n$ has $3$ digits or $4$.

$n$ must have more than $2$ digits because the product of two $2$-digit numbers is not less than $100$.

Similarly $n$ must have fewer than $5$ digits because the product of two $2$-digit numbers is less than $10000$.

This leaves a small enough domain to perform an exhaustive search.

Starting from the bottom:

Hence the cards would be either:
 * $\boxed 3 \boxed 9 \ \boxed 1 \ \boxed 5 \boxed 7$

or:
 * $\boxed 5 \boxed 7 \ \boxed 1 \ \boxed 3 \boxed 9$

It remains to check the rest of the possible numbers.