Professor Stewart's Casebook of Mathematical Mysteries

By the Same Author

Concepts Of Modern Mathematics

Game, Set, And Math

Does God Play Dice?

Another Fine Math You’ve Got Me Into

Fearful Symmetry

Nature’s Numbers

From Here To Infinity

The Magical Maze

Life’s Other Secret

Flatterland

What Shape Is A Snowflake?

The Annotated Flatland

Math Hysteria

The Mayor Of Uglyville’s Dilemma

How To Cut A Cake

Letters To A Young Mathematician

Taming The Infinite (Alternative Title: The Story Of Mathematics)

Why Beauty Is Truth

Cows In The Maze

Mathematics Of Life

Professor Stewart’s Cabinet Of Mathematical Curiosities

Professor Stewart’s Hoard Of Mathematical Treasures

Seventeen Equations That Changed The World (Alternative Title: In Pursuit Of The Unknown)

The Great Mathematical Problems (Alternative Title: Visions Of Infinity)

Symmetry: A Very Short Introduction

Jack Of All Trades (Science Fiction eBook)

with Jack Cohen

The Collapse Of Chaos

Evolving The Alien (Alternative Title: What Does A Martian Look Like?)

Figments Of Reality

Wheelers (Science Fiction)

Heaven (Science Fiction)

The Science Of Discworld Series (With Terry Pratchett & Jack Cohen)

The Science Of Discworld

The Science Of Discworld II: The Globe

The Science Of Discworld III: Darwin’s Watch

The Science Of Discworld IV: Judgement Day

iPad app

Incredible Numbers

Introducing Soames and Watsup

Professor Stewart’s Cabinet of Mathematical Curiosities appeared in 2008, just before Christmas. Readers seemed to like its random mixture of quirky mathematical tricks, games, weird biographies, snippets of strange information, solved and unsolved problems, odd factoids, and the occasional longer and more serious piece on topics such as fractals, topology, and Fermat’s Last Theorem. So in 2009 it was followed by Professor Stewart’s Hoard of Mathematical Treasures, which continued in the same vein with an intermittent pirate theme.

They say that three is a good number for a trilogy. The late Douglas Adams of The Hitchhiker’s Guide to the Galaxy fame did eventually decide that four was better and five better still, but three sounds like a good place to start. So, after a gap of five years, here is Professor Stewart’s Casebook of Mathematical Mysteries. This time, however, there’s a new twist. The short quirky items, such as Hexakosioihexekontahexaphobia, the Thrackle Conjecture, What Shape is an Orange Peel?, the RATS Sequence, and Euclid’s Doodle, are still there. So are more substantial articles about solved and unsolved problems: Pancake Numbers, the Goldbach Conjecture, the Erdős Discrepancy Problem, the Square Peg Conjecture, and the ABC Conjecture. So are the jokes, poems, and anecdotes. Not to mention unusual applications of mathematics to flying geese, clumps of mussels, spotty leopards, and bubbles in Guinness. But these miscellanea are now interspersed with a series of narrative episodes featuring a Victorian detective and his medical sidekick—

I know what you’re thinking. However, I developed the idea a year or so before Benedict Cumberbatch and Martin Freeman’s spectacularly successful modern take on Sir Arthur Conan Doyle’s much-loved characters hit the small screen. (Trust me.) More to the point, it’s not that pair. Not even as portrayed in Sir Arthur’s original stories. Yes, my guys live in the original time period, but across the road at number 222B. From there, they cast envious eyes on the stream of rich clients entering the premises of the more famous duo. And from time to time a case comes up that their illustrious neighbours have shunned or failed to solve: such arcane mysteries as the Sign of One, the Dogs that Fight in the Park, the Catflap of Fear, and the Greek Integrator. Then Hemlock Soames and Dr John Watsup put their brains in gear, show their true colours and their strength of character, and triumph over adversity and lack of market presence.

These are mathematical mysteries, you appreciate. Their solutions demand an interest in mathematics and an ability to think clearly, attributes in which Soames and Watsup are by no means deficient. These passages are signalled by the symbol . Along the way we learn of Watsup’s prior military career in Al-Jebraistan and Soames’s battles with his arch-enemy Professor Mogiarty, inevitably leading to the final fatal confrontation atop the Schtickelbach Falls. And then—

It is fortunate that Dr Watsup recorded so many of their joint investigations in his memoirs and unpublished notes. I am grateful to his descendants Underwood and Verity Watsup for permitting me unprecedented access to family documents, and for generously granting me permission to include extracts here.

Coventry, March 2014

Note on Units

In Soames and Watsup’s era, the standard units of measurement in Britain were imperial, not metric as they mostly are today, and the currency was not decimal. American readers will have no problems with imperial units; admittedly, the gallon is different on either side of the Atlantic, but that unit of measurement doesn’t appear anyway. To avoid inconsistencies I’ve used units appropriate to the Victorian era, even for topics that are not part of the Soames/Watsup canon, except when narrative imperative demands metric.

Here’s a quick guide to the relevant units with metric/decimal equivalents.

Most of the time the actual unit doesn’t matter: you could leave the numbers unchanged, but cross out ‘inch’ or ‘yard’ and replace it by an unspecified ‘unit’. Or choose whichever seems convenient (metre for yard, for example).

Lengths

Weights

Money

Thruppeny bit = colloquial term for a three pence coin.

The Scandal of the Stolen Sovereign

The private detective took his wallet from his pocket, ascertained that it was still empty, and sighed. Standing at the window of his lodgings at number 222B he stared morosely across the street. The strains of an Irish air, expertly played on a Stradivarius, were just discernible above the clatter of passing carriages. Really, the man was insufferable! Soames stared at the stream of people entering the portals of his famous competitor. Most were wealthy members of the upper classes. Those that appeared not to be wealthy members of the upper classes were, with few exceptions, representing wealthy members of the upper classes.

Criminals just weren’t committing the kind of crime that affected the sort of people who would engage the services of Hemlock Soames.

For the past two weeks, Soames had watched with envious eyes as client after client was ushered into the presence of the person they believed to be the greatest detective in the world. Or, at least, in London, which for Victorian England amounted to the same thing. Meanwhile his own doorbell remained mute, the bills piled up, and Mrs Soapsuds was threatening eviction.

He had only one case on the books. Lord Humphshaw-Smattering, owner of the Glitz Hotel, believed that one of his waiters had stolen a gold sovereign: value, one pound sterling. To be fair, Soames could do with a sovereign himself right now. But it was hardly the stuff to attract the sensationalist yellow press, upon whom, deplorable as it might be, his future depended.

Soames studied his case notes. Three friends, Armstrong, Bennett, and Cunningham, had partaken of dinner at the hotel, and had been presented with a bill for £30. Each had given the waiter Manuel ten gold sovereigns. But then the maître d’ noticed that there had been an error, and the bill was actually £25. He gave the waiter five sovereigns to return to the men. Since £5 wasn’t divisible by 3, Manuel suggested that he might keep two of the coins as a tip and give them back one sovereign each, hinting that they were fortunate to have any of the overpayment returned.

The customers agreed, and all was well until the maître d’ noticed an arithmetical discrepancy. Now the men had each paid £9, a total of £27. Manuel had a further £2, making a total of £29.

One pound was missing.

Humphshaw-Smattering was convinced that Manuel had stolen it. Although the evidence was circumstantial, Soames knew that the waiter’s livelihood depended on resolving the mystery. If Manuel were to be dismissed with a bad reference, he would never get another job.

Where did the missing pound go?

See page 249 for the answer.

Number Curiosity*

In detective work, it is vital to be able to spot a pattern. Soames’s unpublished and untitled monograph containing two thousand and forty-one instructive examples of patterns includes the following. Work out

             11 × 91

             11 × 9091

             11 × 909091

             11 × 90909091

             11 × 9090909091

Soames would have used pen and paper, and modern readers may do likewise if they can remember how. A calculator is always an option, but they tend to run out of digits. The pattern continues indefinitely: this can’t be proved using a calculator, but it can be deduced from the old-fashioned method. So, without doing any further calculations, what is

             11 × 9090909090909091?

A harder question is: why does it work?

See page 250 for the answers.

*    Many items in this compilation that do not refer directly to criminal cases are extracted from handwritten notes, some of whose contents have been collected and published, with Soames’s permission, as Doctor Watsup’s Vault of Forensic Anomalies, and will be reproduced without further notification. Some are of later date, added by Watsup’s literary executors, and the assiduous reader will instantly identify such anachronisms.

Track Position

Lionel Penrose invented a variation on traditional mazes: railway mazes. These have junctions like those on railway tracks, and you have to take a route through them that a train could follow, one with no sharp turns. They are a good way to cram a complicated maze into a small space.

Allowed and forbidden routes at junctions

His son, the mathematician Roger Penrose, took the idea further. One of his mazes is carved in stone on the Luppitt Millennium Bench, in Devon, England. That one’s a bit difficult, so here’s a simpler example for you to tackle.

The map overleaf shows the rail network of Tardy Trains. The 10.33 train starts at station S and must finish at station F. The train cannot reverse direction by slowing down and then going backwards, but it can travel along a line in either direction if the track loops back on itself. At points, where two branches join, the train may take any smooth path. What route does the train take?

The maze

See page 252 for the answer, and further information including the Luppitt Millennium Maze.

Soames Meets Watsup

A fine drizzle, of the kind that looks innocuous but quickly soaks you to the skin, was falling on the good citizens of London, and on the bad, as they scurried along Baker Street on errands admirable or nefarious, dodging the puddles. The not-so-famous detective was in his habitual position at the window, staring hopelessly into the gloom, grumbling to himself about his dire finances, and feeling depressed. His incisive solution to the Scandal of the Stolen Sovereign had brought in enough to get Mrs Soapsuds temporarily off his back, but now that the emotional rush of success had subsided, he felt lonely and unappreciated.

Perhaps he needed a like-minded companion? One who could share the daily cut-and-thrust of his personal vendetta against crime, and the intellectual challenge of unravelling the clues that its perpetrators scattered so carelessly across the landscape? But where could he find such a person? He had no idea where to start.

His black mood was interrupted by the appearance of a sturdy figure striding purposefully towards the premises opposite. Instinctively, Soames judged him to be a medical man, recently retired from the army. Well-dressed, well-heeled: yet another wealthy client for that overrated jackass Hol—

But no! The figure inspected the house number, shook his head, and spun on his heels. As he crossed the road, narrowly dodging a hansom cab, the brim of his hat concealed his face, but his body language showed determination, perhaps verging on desperation. Observing the man more closely, now that his interest had been piqued, Soames realised that his coat was not new, as he had first thought. It had been expertly repaired . . . in Old Compton Street, by the look of the stitching. On a Thursday, when the head seamstresses took a half-day off. Down at heel, not well-heeled, he corrected his initial impression, as the man disappeared from view, apparently heading for the doorway below.

A pause: then the bell rang.

Soames waited. A knock at the door announced his long-suffering landlady Mrs Soapsuds, clad in one of her habitual floral print dresses and wearing a large pinafore. “A gentleman to see you, Mr Soames,” she sniffed. “Shall I show him up?”

Soames nodded, and Mrs Soapsuds slouched off down the stairs. A minute later she knocked again, and the medical gentleman entered. Soames waved at her to shut the door and return to her customary place behind the net curtains in her sitting-room on the ground floor, which she did with evident reluctance.

The gentleman listened for a moment, and suddenly tugged the door open, stepping back to allow Mrs Soapsuds to fall sideways to the floor.

“The—uh—mat. Needed dusting,” she explained, picking herself up. Soames silently noted that his landlady also needed dusting, gave her a thin smile, and waved her away. Once more the door closed.

“My card,” the man said.

Soames placed the visiting card face down, unread, and studied the new arrival from head to toe. After a few seconds he said, “Not much of note to identify you.”

“Pardon?”

“Except the obvious, of course. You have been in Al-Jebraistan for the last four years, serving as a surgeon with the Royal Sixth Dragoons. You narrowly escaped a serious wound at the Battle of Q’drat. Your period of service ended soon after, and you decided—after some soul-searching—to return to England, which you did early this year.” Soames peered more closely, and added, “You keep four cats.”

As the man’s jaw dropped, Soames turned over the card. “Dr John Watsup,” he read. “Surgeon, Royal Sixth Dragoons, retired.” His face showed no emotion at this confirmation of his deductions, for it had been inevitable. “Please sit down, sir, and tell me of the crime that has been committed against you. I can assure you that—”

Watsup laughed, a friendly chuckle. “Mr Soames, I am delighted to have met you at last, for your fame has spread far and wide. Your deductions about my person prove that you fully deserve the acclaim that I have encountered. Your modesty at the feat becomes you. But I do not come primarily as a prospective client. Rather, I am seeking a position in your employ. Medicine no longer appeals to me—nor would it to you if you had seen the sights I have been forced to endure at the battlefront. But I am a man of action, I continue to crave excitement, I still have my service revolver, and . . . by the way, how did you do that?”

Soames, ignoring a growing feeling that he was being mistaken for the inhabitant of number 221B, sat down facing Watsup. “By your bearing, sir, I had you marked as a military man before you crossed the road. My eyesight is preternaturally keen, and you have the hands of a surgeon, strong yet lacking the ingrained stains of manual labour. Last December the Times reported that the four-year campaign in Al-Jebraistan was coming to a close and the Royal Sixth Dragoons were returning to England after fighting a decisive but costly battle at Q’drat. You are wearing the appropriate regimental boots, and the wear-patterns on them show you have been back in England for some time. You have a slight scar along your jawbone, almost healed, which was obviously caused by a musket-ball of non-European design—I have written a brief monograph on firearm injuries in the Far East, I must read it to you some time. You are a man of action, as evinced by the way you handled Mrs Soapsuds’s snooping, so you would not have retired from military service voluntarily. If you had been given a dishonourable discharge I would have seen it reported in the scandal sheets, but nothing of the kind has been published recently. Your coat bears four different types of cat hair—not just four colours, which might indicate a single tabby, but different lengths and textures . . . I will spare you a list of their breeds.”

“Astonishing!”

“To be candid, I must also admit that your face is familiar. I am sure that somewhere—ah, yes! I have it! A small article in last week’s Chronicle, with a photograph . . . Dr John Watsup, originator of the well-known phrase ‘Watsup, doc?’ Your fame exceeds my own, Doctor.”

“You are too kind, Mr Soames.”

“No, merely realistic. But if we are to work together, you must convince me that you can think as well as act. Let us see.” And Soames wrote the digits

             4   9

on the back of an envelope. “I want you to insert one standard arithmetical symbol, to produce a whole number between 1 and 9.”

Watsup pursed his lips in concentration. “A plus sign . . . no, 13 is too large. A minus—no the result is negative. Neither multiplication nor division will do. Of course! A square root! Oh, no: , again too large.” He scratched his head. “I am stumped. It is impossible.”

“I assure you there is a solution.”

The silence was broken only by the ticking of a clock on the mantelpiece. Suddenly, Watsup’s face lit up. “I have it!” He picked up the envelope, added a single symbol, and handed it to Soames.

“You pass the first test, Doctor.”

What did Watsup write? See page 252 for the answer.

Geomagic Squares

A magic square is made from numbers, which give the same total along any row, column, or diagonal. Lee Sallows has invented a geometric analogue, the geomagic square. This is a square array of shapes, such that the shapes in any row, column, or diagonal fit together like a jigsaw to make the same overall shape. The pieces can be rotated or reflected if necessary. The left-hand figure shows how this goes; the right-hand one is a puzzle for you to solve. See page 253 for the answer.

Sallows has invented many other geomagic squares, along with generalisations such as a geomagic triangle. See The Mathematical Intelligencer 33 No. 4 (2011) 25–31 and his website

             http://www.GeomagicSquares.com/

Two of Lee Sallows’s geomagic squares. Follow a row, column, or diagonal to find the assembled jigsaw using the corresponding pieces. Left: A completed example. Right: Your task is find the assembled jigsaws for all rows, columns, and diagonals.

What Shape is an Orange Peel?