Henry Ernest Dudeney/Modern Puzzles/72 - Alphabetical Sums/Solution
Modern Puzzles by Henry Ernest Dudeney: $72$
- Alphabetical Sums
RSR ------ PR)MTVVR MVR --- KKV KMD --- MVR MVR ---
Solution
565 ------ 35)19775 175 --- 227 210 --- 175 175 ---
Proof
We have immediately that $R \times R = R \pmod {10}$.
However, because $R \times PR = MTV$ it is clear $R \ne 1$ and (obviously) $R \ne 0$.
So $R = 5$ or $R = 6$.
To get $V$ in the fifth line it is clear that $D = 0$.
Thus we have that $K = 2 M$.
In order for $D$ to be $0$:
- if $R = 5$ then $S$ must be even.
- if $R = 6$ then $S = 5$.
Note that $KMD > MVR$, which implies that $S > R$.
Hence it cannot be the case that $R = 6$ and $S = 5$.
And thus we conclude $R = 5$ and $S = 6$ or $8$.
The divisor $PR$ is a factor of $KMD$, which can only be $210, 420, 630$ or $840$.
Therefore $PR$ is a factor of the lowest common multiple of these numbers, $2520$.
Of the factors of $2520$, only $15$, $35$ and $45$ are $2$-digit numbers ending in $5$.
However $15 \times RSR < 15 \times 600 = 9000 < 10 \, 000$, so $PR \ne 15$.
Finally we check the remaining $4$ possibilities:
- $35 \times 565 = 19 \, 775$
- $45 \times 565 = 25 \, 425$
- $35 \times 585 = 20 \, 475$
- $45 \times 585 = 26 \, 325$
and only $19 \, 775$ matches the pattern $MTVVR$, so it must be our dividend.
The rest of the puzzle can be filled in accordingly.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $72$. -- Alphabetical Sums
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $151$. Alphabetical Sums