The secret code

AN extract from the diary of Dr John H. Watson, MD: I had called upon my friend Sherlock Holmes upon the second morning after Christmas, with the intention of wishing him the compliments of the season. He was lounging upon the sofa in a purple dressing-gown, a pipe-rack within his reach upon the right, and a pile of crumpled papers, evidently newly studied, near at hand. Beside the couch was a wooden chair and on the angle of the back hung a dusty book of mathematics. A lens and a forceps lying upon the seat of the chair suggested that the book had been suspended in this manner for the purpose of examination. "This came from my brother Mycroft," said Holmes, handing me a crumpled paper that I realised was a telegram. I read its contents:

9567
1085
——---
10652

"The sum is correct. However, I do not understand why your brother sent you his grocer’s bill and why should it make you look so worried. These are the only observations I can make." I said. "Then Watson, you have not observed closely," said Holmes and gave me another crumpled piece of paper that had the previous sum written on it in his handwriting. I read out what he had written below it:

"A cipher, a code, a secret language!" I said with the excitement of Archemedes after the discovery of the displacement principle.

He said, "This is an alphametic — a peculiar mathematical puzzle, in which, a set of words is written down in the form of an ordinary long-hand addition sum and it is required that the letters of the alphabet be replaced with decimal digits so that the result is a valid arithmetic sum. For example, there is only one way to replace the letters with digits in the message:

SEND
MORE
——---
MONEY

so that we get the desired and accurate sum. "You can develop your own alphametic secret code, which will be known to you and your associates only. However, there are two rules that every alphametic obeys:

1. The same letter always stands for the same digit and vice versa.

2. The digit zero is not allowed to appear as the left-most digit in any of the addends or the sum.

Also, alphametics are hard to construct. Since we usually deal in base 10, only 10 different letters of the alphabet (at most) can be used. This makes it hard to write phrases or sentences that read well, but make and excellent code. Even if we write down a nice phrase representing a prospective alphametic, the odds that the alphametic will actually be solvable are pretty small. Any truly elegant alphametic should have a unique solution.

The object is to write a phrase or sentence that reads well and assign the values to the letters in the message so that it become a valid alphametic.

ALPH + .A. + METIC is the smallest uniquely-solvable alphametic containing all 10 digits. If you an develop an ordinary code like 9, 1 13, 19 20 21 16 9 4, you are stupid indeed. One can easily deduce that this is a cipher. The first instinct is always to replace digits with the corresponding letters of the alphabet. One can easily deduce that you have written "I am stupid (a=1, b=2 and so on).

However, alphametic codes can be mistaken for a grocer’s bill, rough calculations etc. Since there is only a unique solution, you can find out the message even if you lose your codebook. A letter in different alphametics may stand for different digits, which makes your message harder to decode. For the person who decodes your cipher, the message is elegant to read, since it is always short and poetic."

"Splendid Holmes," I said. "Elementary, my dear Watson," said Holmes. "I am glad you said that at last," I replied.

— Aditya Rishi