Upon that memorable afternoon getting on thirty years before, Tom and his grandfather had swiftly reckoned up the digits in the answers to the rest of the multiplication tables. He remembered the excitement that he’d felt, the giddy, sheer thrill of discovery now come back to him in a fruitcake rush of allspice, candied peel and Snowy Vernall’s rubbing liniment. The two-times-table, if you added up the figures of the products, it transpired, resulted in a number-pattern that first ran through all the even numbers, two, four, six, eight, then all of the odd ones, one (one plus nought), three (one plus two), five, and so on up to nine (eighteen, or one plus eight). Remembering the way the one- and eight-times-tables had both yielded up numerical progressions that were backwards mirror-versions of each other, Snowy and young Tom had looked at the seven-times-table, where they’d learned that first the added answers counted down through the odd numbers, seven, five (one plus four), three (two plus one), one (two plus eight, which made the ten, the digits of which added up to one), and then went on to run down through the even numbers. Eight (three plus five), six (four plus two) and so on until the countdown of odd numbers started up again. The number seven seemed to work exactly like the number two, but with the sequence running back to front.
The number three, which just went three, six, nine, three, six, nine, unendingly if you made sums out of its multiples, appeared to be twinned with the number six, which went six, three, nine, six, three, nine, if you did the same thing. The number four produced a pattern that seemed complicated at first sight, in that it counted down from two numbers in parallel, and alternated in between the two. Thus, what you got was four, then eight, then three (or one plus two), then seven (one plus six), then two (two plus nought), six (two plus four), one (two plus eight, adding up to ten, or one plus nought), five (thirty-two, or three plus two), et cetera, et cetera. The five-times-table, unsurprisingly by now, did just the same thing in reverse. It alternated in the same way between two progressions, this time counting up instead of down, so that the sequence in this case was five, one, six, two, seven (two plus five), three (three plus nought), eight (three plus five) and so forth. Tommy and his grandfather had looked at one another and just burst out laughing so that Tom’s grandma Louisa had come out the kitchen to see what was up.
What had been up was that there seemed to be a hidden pattern in the sums that could be generated by the answers of the one- to eight-times-tables. They were all symmetrical, one mirrored eight, two mirrored seven, three worked just the same as six, four was like five. Only the number that had sparked off their investigations, nine, remained alone out of the single figures in that it did not possess a twin, a number that no matter how much it was multiplied would yield the same unvarying result.
Tom, eight years old, had been attempting to explain all this to his uncomprehending gran, when out of nowhere his granddad had yelped with glee, snatched up the midget pencil and, in faint lines on the thin and shiny butcher’s paper littering the table, had inscribed two circles, one inside the other. With one Capstan-yellowed index finger, Snowy had jabbed meaningfully at the drawing, looking up at Tommy from beneath the winter hedgerow of his brow to ascertain whether his grandson understood or not. The old man’s eyes were shining in a way that had reminded Tommy, there amidst the fruited oven-fug and camaraderie of the maths game which they’d been working out together, that his grandfather was said by many to be mad, including Tommy’s mam. And everybody else, now that he’d thought of it. His granddad had just grinned and once again poked at his mystifying scribble with an urgent finger. All that there had been to Snowy’s drawing was just two concentric circles, like a car tyre, or an angel’s halo standing on its side. Tommy had squinted at the simple shape for what seemed minutes before he’d become aware that he was looking at the figure nought.
It had been just as if the lights had been switched on. Nought was the only number, other than the number nine, that didn’t change if it were multiplied. All of the single digit figures between nought and nine made sequences by adding up their multiples that had a perfect symmetry. As if to underline this, Tom’s granddad had once more taken up his pencil, and had written those ten numbers in, all in a ring between the zero’s innermost and outer circles, like the numbers round the edges of a clock. The number nought was roughly where the one would be upon a normal timepiece, with the numerals proceeding clockwise round the dial and leaving spaces where the six and twelve were usually positioned. The effect of this was that each number was now set at the same horizontal level as its mirror-twin, the nine up at the top left face now lined up with the nought at the top right. The eight and one were opposite each other at both ten-to and ten-past, the seven and the two were diametrically opposed, each at the quarter-hour mark, with the six and three below that, and the five and four facing each other down the bottom, one at five-and-twenty-to, the other one at five-and-twenty after. It was lovely. In one simple flash a hidden pattern that had been there all the time, concealed beneath the surface, was revealed.
Neither Tom nor his grandfather had had the first idea what their discovery might mean, or could conceive of any useful application for it. Indeed, it was so blindingly obvious once you’d first seen it that they’d both assumed that someone, or more likely a great many people, had stumbled across the notion previously. It didn’t matter. In that moment Tom had felt a sense of triumph and sultana-scented revelation that he’d never known before or since. His grandfather had smiled a cracked smile that looked rueful rather than elated, and had stabbed once more with one black fingernail at the blank space enclosed by the big number nought’s interior ring.
“The nought’s a torus. That means, like, a lifebelt shape what’s got a hole in. Or it’s like a chimneypot, looked down on from above. And at the middle of the nought here, down the barrel of the chimney, that’s where all the nothing’s kept. You’ve got to keep your eye on nothing, lad, or else it gets all over everything. Then there’s no chimneypot, there’s just the hole. Then there’s no lifebelt, there’s no torus. There’s no nothing.”
With this, Snowy Vernall had seemed to get angry or unhappy, just like that. He’d screwed the piece of paper with the altered clock face drawn upon it up into a ball and thrown it on the fire. Tom hadn’t comprehended any of what his granddad had just been going on about, and must have looked scared by the old boy’s sudden change of disposition. Tommy’s gran Louisa, who looked like she’d seen these swings of mood before, had said “Right, that’s enough sums for today. Young Tommy, you run off back home before your mam gets worried. You can see your granddad Snowy on another Sat’day afternoon.” She hadn’t even shown Tom out, perhaps because she’d known that there was an explosion imminent. Tommy had barely shut the worn front door behind him and stepped outside into Fort Street when he heard the furious bellowing and, shortly after, breaking glass. Most probably it would have been a window or a mirror, mirrors being something that Tom’s grandfather was known to have become suspicious of. Tommy had scarpered off down Fort Street which, although it had been barely the mid-afternoon, Tom pictured now as having then been ominously dark. However, he recalled that this had happened in the ’Twenties, long before the Borough Waste Destructor had been pulled down to make way for flats in Bath Street, so that was one mystery solved.