Henry Ernest Dudeney/Puzzles and Curious Problems/95 - Beeswax/Solution

by : $95$

 * Beeswax

Solution
The key is:

1 2 3 4 5 6 7 8 9 0 --- A T Q B K X S W E P

and the solution to the subtraction is:

9 1 7 9 4 7 4 7 6 - 4 0 8 8 5 7 9 2 3 --- 5 0 9 0 8 9 5 5 3

So BEESWAX represents the number $4 \, 997 \, 816$.

Proof
From the thousands place we have:
 * "$S - S = E$"

If there was no borrow from the hundreds place we will have $E = 0$.

But the first number begins with $E$, so $E \ne 0$.

Thus there must be a borrow from the hundreds place, giving $E = 9$.

Now we look at the hundreds place.

If there was a borrow from the tens place we will have:
 * $10 + B - 1 - E = B = K$

which is impossible since all letters represent distinct digits.

Thus there is no borrow from the tens place, and
 * $10 + B - E = B + 1 = K$

In the ten-thousands place we have:
 * "$W + K = B$"

Since $B < K$ and there is a borrow from the thousands place, we have instead:
 * $W + K = 10 + B - 1 = 8 + K$

giving $W = 8$.

For the hundred-thousands place we have a borrow from the ten-thousands place.

Therefore we have:
 * $P + W = E - 1$

Since $W = 8$ and $E = 9$, this gives $P = 0$.

Combining the millions place and ten-millions place, we have, with no borrows:
 * $E + W = 10 A + S$

so we have $A = 1$ and $S = 7$.

The hundred-millions place gives:
 * $B + K = E = 9$

Since we have $B + 1 = K$ from earlier, we can conclude:
 * $B = 4, K = 5$

We have so far: 9 1 7 9 4 7 4 7 X - 4 0 8 8 5 7 9 T Q --- 5 0 9 0 8 9 5 5 Q

Since the remaining untaken digits are $2, 3$ and $6$, we must have:
 * $T = 2, Q = 3, X = 6$

and thus BEESWAX represents $4 \, 997 \, 816$.