Hidden Figures

This is not a film review but a short introduction to verbal arithmetic. A word sum, or cryptarithm, is an equation, often involving a sum of two numbers, where the digits have been consistently replaced by letters. The object of the puzzle is to recover the digits from the pattern of the letters and the constraints of the equation.  Here is a famous example, which was published in the July 1924 issue of Strand Magazine by the mathematician Henry Dudeney, who had a great talent for devising interesting puzzles.

                        S E N D

                    +  M O R E

                     M O N E Y

So the solver must assign the digits 0—9 to the letters so that the equation holds true. It turns out there is only one solution to SEND + MORE = MONEY, and this is O = 0, M = 1, Y = 2, E = 5, N = 6, D = 7, R = 8, and S = 9.

Word sums usually satisfy three conditions:

1. The words used must be real and should involve at most (preferably exactly) 10 different letters.
2. The numbers may not begin with 0 (in the above example S and M are not zero).
3. Subject to conditions 1 and 2, the solution should be unique.

It is a bonus if the words form a meaningful phrase or have some obvious association, and SEND MORE MONEY satisfies all these requirements.

How about this one?

                        R E A D

                       +     M Y

                        B L O G

It has exactly 10 different letters, so the first requirement is satisfied. The second is OK if we rule out zero for R, M and B. It is the third requirement that fails because there are altogether 36 distinct solutions, which of course makes it easier to find one. Have a go!

However, I can narrow it down by asking that MY should be a prime number whose digits add up to 7. In fact, with this proviso, there is only one solution. Have another go!

A solution is determined entirely by the pattern of the letters and the arithmetic. So solutions to READ + MY = BLOG also yield solutions to HAND + IT = OVER, but not to GIVE + IT = BACK.

The equation can have a different format, as in MIND = THE + GAP. Unfortunately, this fails Condition 3 badly with 96 distinct solutions. Notice, however, that for any given solution, we can swap the numbers THE and GAP to get another one, so there are really only 48 ‘essentially different’ solutions.

The equation can involve different arithmetical operations. I used to work in a student support team called sigma and I set this challenge using multiplication instead of addition: HELP × ME = SIGMA. This has only one solution with non-zero digits.

The same idea can be applied to a narrative rather than an equation. For example, this appeared as a Sunday Times Teaser in 2016:

Which digits, consistently replacing the letters, make all these statements true?

                        TRIPLE is a multiple of three.

                        EIGHT is a cube.

                        NINE is divisible by nine.

                        PRIME is a prime.

Finally, since Christmas is just around the corner, here is a seasonal challenge:

XMAS + PUDD = TASTY.

There are only 9 letters, so let’s rule out the digit 7. In this case, it is your job to find the number TASTY, which is unique.

(I will include solutions at the end of my next post.)