“But what,” I pressed, “is three-dimensional chess? Do you play it in a cube?”
“I have no idea,” retorted Simon, becoming flustered. “I don’t want to think about it just now.”
“I mean, if you play two-dimensional chess in three-dimensional space, does that mean three-dimensional chess has to be played in four-dimensional space?”
“Aaaah, hhnnnn…uuugh…”
“There’s a mention here on the Internet that it’s played in Star Trek. They seem to be using one of those cake stands you find in village tea shops. Was yours played on a cake stand?”
“Hnnnn,” he gave a Grunt Number Four (frustrated) and eyed a savaged Jiffy bag. He dropped the wreckage onto the pile next to his right knee, ready to be gathered into the green wheeliebin, picked up an envelope from a thermal underwear company and assaulted the plastic window.
“Please be quiet. I have to concentrate now.”
Always, in these disembowelings, he takes care to include a windowectomy, and set that non-recyclable material aside, ready for transport to his black wheeliebin.
Goodness gracious, where was Simon when Simon was about? Here it is, in the moldy yellow Eton box file I took from the back room of the Excavation: the rules and board for three-dimensional chess. Simon not only played the game, he invented it.
* * *
1 From the Manchester Evening News, November 3, 2007:
A LOTTERY scratchcard has been withdrawn from sale by Camelot—because players couldn’t understand it. The Cool Cash game—launched on Monday—was taken out of shops yesterday after some players failed to grasp whether or not they had won. To qualify for a prize, users had to scratch away a window to reveal a temperature lower than the figure displayed on each card. As the game had a winter theme, the temperature was usually below freezing.
But the concept of comparing negative numbers proved too difficult for some…Tina Farrell, from Levenshulme, called Camelot after failing to win with several cards. The 23-year-old, who said she had left school without a maths GCSE, said: “On one of my cards it said I had to find temperatures lower than −8. The numbers I uncovered were –6 and −7 so I thought I had won, and so did the woman in the shop. But when she scanned the card the machine said I hadn’t.
“I phoned Camelot and they fobbed me off with some story that –6 is higher—not lower—than −8 but I’m not having it.”
*21 3D chess
It’s NOT 3D chess, Alex, please try to get this right: it’s three-player chess.
Simon
* * *
(as invented by Simon)
1. Pieces laid out in Fig 1. Squares in Fig 2. Green has his pieces at the bottom; his left-hand opponent is Black, his right-hand opponent Orange.
2. Each person has two kings. One, the left king, may be checked only by the left-hand opponent, and the other, the right king, only by the right-hand opponent. The kings are marked with a cross according to the color of the opponent who is allowed to check it.
3. There are nine kinds of men:
a. 2 kings, cuboids with crosses, marked as above (K)
b. 1 queen, a cuboid with a circle on top (Q)
c. 1 general, a cuboid with a triangle on top (G)
d. 2 rooks, cylinders (R)
e. 3 bishops, cubes (B)
f. 2 ordinary knights, paper clips in cardboard bases (KT)
g. 2 gunner knights, triangular prisms (faces missing) with circles on top (Ng)
h. 2 anti-gunner knights, like gunner knights except for dots on top (Nag)
i. 13 pawns, tetrahedral
4. Captures are made as in chess, by moving onto a square occupied by the piece. The pawn is an exception that will be dealt with later.
5. The moves of the pieces are as follows:
a. The bishop moves parallel to the edges of its rhombus, so as to stay on squares of a certain color, e.g., B(GG5)—DE3.
b. The rook moves perpendicular to the edges of its rhombus, e.g., R(GH6)—BG3.
c. The queen moves like the rook or bishop.
d. The king1 moves like the queen, except only one rhombus at a time.
e. The gunner knight moves one rhombus B-wise and one rhombus R-wise, to make a move that cannot be done by the Q e.g. Ng(OG6)—GF6.
f. The anti-gunner knight moves like the gunner knight, reversed.
g. The ordinary knight moves like one of the two preceding knights.
h. The general either moves like an ordinary knight or moves along its long diagonals, e.g., G(BD6)—BD2.
6. Pawns, when on rhombi pointing to their possessor, move directly forward one rhombus. Take R-wise left or right forward. When on other rhombi, they move directly (R-wise) forward one, two rhombi, or three first time. They take R-wise or B-wise left or right forward. If they move two, three rhombi, they may be taken by a pawn as if it had moved only one rhombus. All pawns may be promoted to any piece (except K), on reaching rhombi from which they cannot move. E.g., (for green ones) P(GF3)—GF4, P(0D2)—BH4, P(BF3)—BE2, P(GE5)xQ(BE3), P(0G4)xR(GE5), P(BD5)xK(GG6), P-GE7 = Q.
* * *
1 It is possible to exchange the two kings if both are in check.
* * *
The game is based on that hydrocarbon shape Simon purified in his notebook. It has re-emerged, bubbled back up to the surface from his mathematical depths, confabulated, fluted, overlaid with letters and recaptured numbers, Sellotaped to a sheet of lined file paper, and ready for use in your tutor’s pipe-smoke-filled room.
The top half of the illustration on page 184 is the playboard. As an ordinary chessboard has black squares and white squares, a three-dimensional chessboard (for three players) has black diamonds, white diamonds, and diamonds labeled “A.”
The bottom half gives the location reference for each diamond.
Here’s a tidier version of the playboard, with the diamonds labeled “A” replaced by gray-shaded diamonds:
Each of the three players has:
Two Kings (K) , one Queen (Q) , one General (G) , two Rooks (R) , three Bishops (B) , two ordinary Knights (Kt) (represented by “paper clips in cardboard bases”), two Gunner Knights (Ng) , two Anti- gunner Knights (Nag) and thirteen Pawns (P) .
*22 Breakthrough
* * *
Hurry! Hurry! Back to the classroom! Time for more misdeeds with Triangle and Square!
To recap: Square and Triangle are symmetrical. You can do all manner of things to them—rotate them, flip them over, in any combination you fancy—and they’ll always look just the same after your wickedness as before:
In short, Group Theory is the study of the moves that always leave an equilateral triangle and a square (or other symmetrical object) looking the same. To show how these different moves combine (e.g., one turn of a square, followed by one more, equals two turns in total), we use a sudoku-type grid called a Group Table…Oi! You, yawning at the back! Pay attention!
* * *
How does a subject that’s for children—cutting squares and triangles out of dayglo paper, representing them being bullied and shoved about by symbols such as and lead to an intellectual discipline so intricate and metaphysically profound that it can absorb a genius like Simon for the majority of his adult life? These Group Tables for the turns and twists of Square and Triangle seem too bland for discussion. Their straightforwardness as secretarial devices makes them appear beyond analysis, stagnant with simplicity. How could there possibly be anything extra to say about them?
But there is.
The mathematics libraries of the world are crammed to the ventilation ducts with books that depend on an easy idea, gently hidden inside the Group Table for the rotations of Square but missing from the Table for Triangle.
It’s called a “Subgroup.”
Square’s Group Table of turns has one, and Triangle’s doesn’t.
(There’s a slight fuss about what Simon calls “trivial” Subgroups, which all groups possess, but we’ll ignore this complaint here.)
A Subgro
up is simply a Group within a Group, a smaller symmetry hidden inside the larger one. The Group Table for the turns of Triangle doesn’t contain any Subgroups. That’s why it’s considered an “atom” of symmetry. It can’t be broken down into smaller symmetries.
A square’s Group Table of rotations does contain a Subgroup. It can be further broken down in symmetry terms so, therefore, it is not an “atom” of symmetry but a flibbertigibbet.
Subgroups are one of the keys to turning Groups from a distraction for children into a mathematical smash-and-grab raid for Simon Nortons.
In Chapter 25, we’ll see how to spot the hidden Subgroup in the Group Table of Square.
A last paragraph, for advanced readers: the “atoms” of symmetry are to Group Theory what the prime numbers are to whole numbers—the building blocks of the entire system. Just as any whole, positive number can be constructed by multiplying together prime numbers, any finite Group can be made up by combining together these “atoms” of symmetry. For example, the number 15 is composed by multiplying together the two primes 3 and 5. Similarly, the rotations of a pentadecagon (a fifteen-sided regular shape) can be constructed by combining the rotations of a triangle and the rotations of a pentagon.
23 Breakdown!
I don’t feel that the episode of the toilet falling through the floor has anything to do with my character.
Simon
PING! An email, from Simon:
Friday, October 22nd, six minutes before midnight
Subject: detailed comments on your book
Let me start by explaining about Pasquino and Nick. You don’t know about Pasquino. He was an Ashdown boy. At one stage he and someone called Carpenter were my best friends there—well it might be more accurate to say my only friends. Then one day they joined the herd of bullies. I may be able to forgive Malcolm who, after all, was bullied himself, but I will never be able to forget Pasquino and Carpenter.
Nick is another traitor, this time not just against me but against most of the country. You do know Nick—Nick Clegg. Thanks to the Cameron Cuts he’s signed up for, many of the bus and train trips I’ve been doing recently, and even more that I haven’t got round to doing, may no longer be possible.
Now let me remind you of your promise that I would be allowed to vet all material for publication. I accept that I cannot have a 100% veto but the general tenor of what you have written is so far from acceptable that if you go ahead with it in anything like its present form you will have broken every one of the promises you made and will thoroughly deserve the description of traitor.
It was with difficulty that you managed to persuade me to include a description of my rooms; you said that without that any biography would be incomplete. I reluctantly agreed on the premise that you would do your best to tone it down and that you would help me tidy it up so that by the time the book appeared I would have nothing to fear from any possible visit by housing inspectors.
Furthermore you said that I could use the book as a soapbox for the issues on which I care deeply.
So what has happened? Far from toning it down, you have done your best to emphasise the issue by putting the words which you say I won’t allow you to use, but which an astute reader will be able to deduce, in display format. You return to this again and again as if to make sure that even the dimmest reader is aware of how awful my rooms are. You use the insulting word “excavation” repeatedly. You explicitly mention my fear of the housing inspectors thus ensuring that any of them who read the book will know exactly where to go to find something wrong; and as your interpretation of helping me to tidy up consists of doing perhaps 0.1% of the work and expecting me to do the rest all alone, there’s little chance that I will be able to rectify things before they arrive.
And to cap it all, you confine all mention of my campaigning activities to the barest minimum, in spite of my repeated statements that they are essential for understanding my life (e.g. they account for my loss of interest in maths)! Not so much concern with completeness now, eh?
Thinking about my life I can detect four main turning points.
1966: Start work at London University: mentioned
Leave confining atmosphere of Eton: understated
Find cache of London Transport publicity: mentioned but with facts completely wrong.
1969: Start work at Cambridge: mentioned
Meet Professor Conway: mentioned
1985: Lose Professor Conway to Princeton: mentioned but understated
Bus cuts culminate in chaos of deregulation: ignored, although this is behind both my move towards public transport campaigning and the untidiness of my rooms (I couldn’t keep track of all the timetable changes and eventually gave up trying).
2002: Mother dies leading to my purchase of a flat in London: ignored
I start to feel my age: ignored.
So you have either ignored or understated the majority of the key events in my life.
As I’m sure I’ve said, it was NOT going on trips that led me to lose interest in maths, it was getting involved in campaigning after the deregulation of bus services. Conway moving to Princeton also deprived me of the stimulus behind much of the maths I did.
I don’t think there’s any hope that your next draft will be fully acceptable, but I do think that if you take account of everything I say then only minor changes will be needed.
Now let me turn to the subject of truth. I think I told you how my enjoyment of one of Victor Canning’s books, The Finger of Saturn, was completely spoilt when he referred to a railway viaduct as “long disused.” It was on the Beeching hitlist but it survived and is today marketed as one of the major scenic routes of Devon and Cornwall. The idea that it could have been closed depressed me profoundly and spoilt my enjoyment of the book. So, while I accept the concept of writer’s licence, I see no reason for many of the factual errors you have included, such as confusing Sharyn McCrumb’s Bimbos of the Death Sun with her If I’d Killed Him When I Met Him.
Of course, if you prefer to get things wrong deliberately, as what you say on page 35 might suggest, you belong on the team of a trash publication like the National Enquirer as I said in my last message.
Now let me start on the detailed critique…
[There followed five pages of corrections.]
24
I used to worry at school that there was I enjoying numbers when everyone else was made to do boring things, like swimming or the American Civil War. Why didn’t the teacher tell them about the cyclic permutations of 142857?
Simon
“No, don’t lie on the floor, please, Alex—do you mind if I call you Alex?”
It was the hypnotherapist speaking. Andrew Cunningham: a slight, soft-spoken man, the star of dozens of TV shows for curing stage fright, fear of dogs and heights, and for cramming the tremulous with braggadocio. For the Channel 4 program Faking It, he transformed a city solicitor into a foul-mouthed, shaven-haired garage music DJ.
I was going to be hypnotized. Hypnosis, I’d decided, would help me investigate the curlicues of Simon’s mind.
One Harley Street is the last chance a hypnotist has to work in this prestigious part of London. It is the high-end numbers of Harley Street that belong to the lucrative professions like sucking out fat with a vacuum cleaner and burning women’s faces with acid. Fail to fit into some cubbyhole at Number 1, and you drop off into Cavendish Square; a little farther south, and before you know it your business is camping alongside Clairvoyant Claire under Vauxhall Bridge.
I’d met Andrew in the entrance hall of this office block, beside a brass plaque stretched thin with company names suggesting empty one-room offices and unmanned telephones: Mindworks, Mindspa Clinics, Management Psychology Ltd, The London Therapy Center. We had to go up and down so many corridors and back staircases to get to Andrew’s consulting room that possibly it was no longer in London at all.
“The first thing to know about hypnotism is that it doesn’t exist,” Andrew said with satisfaction. “Here we
are, third door on the left.” After the urn of potpourri.
My idea was that if you could identify what motivated a monomaniacal genius like Simon to do mathematics all day long, then ask a hypnotist to make you feel that motivation too, you won’t be suddenly good at math, but you might begin to study it in the correct pioneering spirit and force yourself to be better.
The noise of traffic outside disturbed his curtains.
“Hypnotism can’t add something you haven’t already got,” Andrew insisted. He’s had success helping people to flush their cigarettes down the toilet because “Anyone can give up. The potential is already there. You’re not making them do something they can’t.” Hypnosis “removes blocks to what is natural.” It’s similar to extreme daydreaming. “If you talk to people who’ve been hypnotized, they won’t say, ‘He clicked his fingers and I was under, then I woke up and I don’t know what happened.’ They’ll say, ‘I was aware, I just felt like pretending I was an artichoke.’” You can mesmerize people to eat lemons believing they’re peaches, or to see green when they’re staring at purple, and people really do see green. Clinical psychologists have done experiments. The part of the brain that sees green registers. Everyone’s got the ability to be fooled.
Hypnotism is about “rapport: get the subject to feel responsive, then make suggestions that are easily taken up.”
“Such as, ‘Congratulations, you’re a maths genius’?”