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.)

The Loney

I have joined a reading group in Stratford upon Avon. It’s called Books with Friends, and its 45 members are divided into three smaller groups that meet at different times on the last Thursday of each month. And …, thanks to firm but good-natured guidance from our chair, who attends all 3 sessions,  we spend most of our allotted hour and a quarter ACTUALLY DISCUSSING THE MONTHLY BOOK !

This month’s book is The Loney by Andrew Michael Hurley. It was published in late 2014 and in 2016 won the “Best Book of the Year” British Book Industry award, as well as the Costa First Novel award. Although this was his first full-length novel, Hurley had previously published two collections of short stories.

The novel is about a Catholic family from London and their teenage son Andrew (Hanny), who has not spoken since birth. His younger brother, who is nicknamed Toto and tells the story, is very close to Hanny and understands the language of objects Hanny uses to communicate his feelings.  The central narrative describes a visit the Smith family makes to an isolated house (The Moorings) on a wild and remote stretch of Cumbrian coast called “The Loney”. They are accompanied by some close friends and their local priest. The visit is a religious retreat and a pilgrimage organised by Hanny’s strong-willed mother, who desperately hopes her son will be ‘cured’ by taking the waters at a nearby shrine on Easter Monday. The novel’s setting is a familiar one, especially favoured by crime writers: a small group of people thrown together in enclosed and isolated surroundings facing uncertainty, unease and perhaps danger.

The Loney itself, and the local inhabitants, play a big part in establishing a brooding atmosphere of menace, superstition and hostile natural forces. A second focus of the action is a sinister house called Thessaly on Coldbarrow, a spit of land twice a day cut off by the sea and separated from the marshes around The Moorings by a treacherous stretch of beach with unpredictable tides and dangerous currents. The locals are a motley bunch, portrayed at once as superstitious, threatening, knowing, and somewhat dysfunctional. There are unpleasant goings-on during the Smith’s visit: an effigy of Christ is strung up at night in a nearby wood with a pig’s heart inside, vandalism at a local Church stops the family celebrating Easter mass, a mangled body of a new-born lamb is found near the house.

The book is beautifully written. The author has the knack of bringing scenes vividly to life by his observations of everyday objects, sounds and smells. He is also good at conveying his characters’ personalities through their dialogue and their actions and reactions. You gradually learn about them by what they say and do. The tender relationship between Hanny and Toto is sensitively explored, and the religious and philosophical musings of father Bernard, the priest, are cleverly woven into the plot and ring true. His predecessor in the family’s London parish, father Wilfred, had preached church dogma and moral certainties, sadistically chastising signs of human frailty in the young. But we learn that he apparently took his own life when he could no longer sustain the fiction of his own faith. By contrast, Father Bernard faces up to his moral dilemmas and works hard to fulfil his duty of love to the members of his flock, finding consolation in his hipflask when the going gets tough.

The dramatic build-up to the pivotal moment when Hanny is taken to the sacred well is shocking, but well controlled and convincing. Faith and prayer are not enough; miracles no longer happen. But sadly, from this point on, the novel descends into poorly-plotted melodrama. Having so painstakingly exposed the false hopes of a simplistic view of religion and the worthless points on its spiritual loyalty cards, the author then hands unbelievable miraculous powers to the opposition. Satan may often have the best lines but he surely doesn’t deserve a monopoly on instant magical cures. I could probably have coped with some subtle hints of black magic at large in the primaeval hinterland of The Loney, provided that they had been sufficiently ambiguous. But here the shenanigans in Thessaly’s dark cellar involving a sacrificial baby and a back story of a woman once hanged nearby for witchcraft more than just strain credibility, they descend into farce. At this point the short-story writer clearly felt that he had gone on too long and needed to round things quickly off. Fast forward of 20 years: Hanny is now happily married with his human faculties fully restored, while his brother Toto is having therapy to cope with no longer being needed. Where the absence of explanation worked so successfully in conjuring up the menace earlier, the same technique utterly fails to make the necromantic events on Coldbarrow, and the motivations of the people involved, even faintly believable, and my imagination certainly couldn’t make good the deficiency.

At Books with Friends we are each expected to give the book we have just been reading a score out of 10. Half way through The Loney, I was all set to award an 8 or 9, which are high scores for me, but by the end I was sadly reduced to a 3 or 4. A real let-down after such a good start.