Henry Ernest Dudeney/Puzzles and Curious Problems/29 - The Shopkeeper's Puzzle/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $29$

A shopkeeper uses a code word where each letter stands for the digits from $0$ to $9$.

What is the code used to encode this addition sum?

  GAUNT
+ OILER
 ------
 RGUOEI

REGULATION

As this is an addition sum with a $6$ digit sum, the first digit of the sum is $1$.

Hence $R = 1$.

Because $N + E = E$, either $N = 0$ or $N = 9$ and there was a carry from the right.

But the only way there can be a carry from the right is from $T + R = I$, where $R = 1$.

Hence $T$ would have to be $9$.

But that would make both $T = 9$ and $N = 9$, which cannot happen.

Hence $N = 0$.

We have that $G + O + 1 = G$, so $O = 9$.

Thus we have so far:

 1 2 3 4 5 6 7 8 9 0
--------------------
 R               O N

and it is apparent that the only anagram of GAUNT OILER with this pattern is REGULATION.

Hence we have:

 1 2 3 4 5 6 7 8 9 0
--------------------
 R E G U L A T I O N

and the completed cryptarithm:

  36407
+ 98521
 ------
 134928

$\blacksquare$