Childe raised his hand above the right-side shape. ‘I’m prepared to press it.’
‘Wait.’ I climbed the steps and walked over the threshold, joining Childe. ‘I don’t think you should be in here alone.’
He looked at me with something resembling gratitude. None of the others had stepped over yet, and I wondered if I would have done so had Childe and I not been old friends.
‘Go ahead and press it,’ I said. ‘Even if we get it wrong, the punishment’s not likely to be too severe at this stage.’
He nodded and palmed the right-side symbol.
Nothing happened.
‘Maybe the left side… ?’
‘Try it. It can’t hurt. We’ve obviously done something wrong already.’
Childe moved over and palmed the other symbol on the top row.
Nothing.
I gritted my teeth. ‘All right. Might as well try one of the ones we definitely know is wrong. Are you ready for that?’
He glanced at me and nodded. ‘I didn’t go to the hassle of bringing in Forqueray just for the free ride, you know. These suits are built to take a lot of crap.’
‘Even alien crap?’
‘About to find out, aren’t we?’
He moved to palm one of the lower symmetry pairs.
I braced myself, unsure what to expect when we made a deliberate error, wondering if the Spire’s punishment code would even apply in such a case. After all, what was clearly the correct choice had elicited no response, so what was the sense in being penalised for making the wrong one?
He palmed the shape; still nothing happened.
‘Wait,’ Celestine said, joining us. ‘I’ve had an idea. Maybe it won’t respond — positively or negatively — until we’re all in the same room.’
‘Only one way to find out,’ Hirz said, joining her.
Forqueray and Trintignant followed.
When the last of them had crossed the threshold, the rear door — the one we had all come through — slid shut. There were no markings on it, but nothing that Forqueray did made it open again.
Which, I supposed, made a kind of sense. We had committed to accepting the next challenge now; the time for dignified retreats had passed. The thought was not a pleasant one. This room was smaller than the last one, and the environment was suddenly a lot more claustrophobic.
We were standing almost shoulder to shoulder.
‘You know, I think the first chamber was just a warm-up,’ Celestine said. ‘This is where it starts getting more serious.’
‘Just press the fucking thing,’ Hirz said.
Childe did as he was told. As before, there was an uncomfortable pause which probably lasted only half a second, but which felt abyssally longer, as if our fates were being weighed by distant judicial machinery. Then thumps and vibrations signalled the opening of the door.
Simultaneously, the door behind us had opened again. The route out of the Spire was now clear again.
‘Forqueray…’ Childe said.
The Ultra tossed the float-cam into the darkness.
‘Well?’
‘This is getting a tiny bit monotonous. Another chamber, another door, another set of markings.’
‘No booby-traps?’
‘Nothing the drone can resolve, which I’m afraid isn’t saying much.’
‘I’ll go in this time,’ Celestine said. ‘No one follow me until I’ve checked out the problem, understood?’
‘Fine by me,’ Hirz said, peering back at the escape route.
Celestine stepped into the darkness.
I decided that I was no longer enjoying the illusion of seeing everyone as if we were not wearing suits — we all looked far too vulnerable, suddenly — and ordered my own to stop editing my visual field to that extent. The transition was smooth; suits formed around us like thickening auras. Only the helmet parts remained semi-transparent, so that I could still identify who was who without cumbersome visual tags.
‘It’s another mathematical puzzle,’ Celestine said. ‘Still fairly simple. We’re not really being stretched yet.’
‘Yeah, well, I’ll settle for not being really stretched,’ Hirz said.
Childe looked unimpressed. ‘Are you certain of the answer?’
‘Trust me,’ Celestine said. ‘It’s perfectly safe to enter.’
This time the markings looked more complicated, and at first I feared that Celestine had been over-confident.
On the left-hand side of the door — extending the height of the frame — was a vertical strip marked by many equally spaced horizontal grooves, in the manner of a ruler. But some of the cleanly cut grooves were deeper than the others. On the other side of the door was a similar ruler, but with a different arrangement of deeper grooves, not lining up with any of those on the right.
I stared at the frame for several seconds, thinking the solution would click into my mind; willing myself back into the problem-solving mode that had once seemed so natural. But the pattern of grooves refused to snap into any neat mathematical order.
I looked at Childe, seeing no greater comprehension in his face.
‘Don’t you see it?’ Celestine said.
‘Not quite,’ I said.
‘There are ninety-one grooves, Richard.’ She spoke with the tone of a teacher who had begun to lose patience with a tardy pupil. ‘Now counting from the bottom, the following grooves are deeper than the rest: the third, the sixth, the tenth, the fifteenth… shall I continue?’
‘I think you’d better,’ Childe said.
‘There are seven other deep grooves, concluding with the ninety-first. You must see it now, surely. Think geometrically.’
‘I am,’ I said testily.
‘Tell us, Celestine,’ Childe said, between what was obviously gritted teeth.
She sighed. ‘They’re triangular numbers.’
‘Fine,’ Childe said. ‘But I’m not sure I know what a triangular number is.’
Celestine glanced at the ceiling for a moment, as if seeking inspiration. ‘Look. Think of a dot, will you?’
‘I’m thinking,’ Childe said.
‘Now surround that dot by six neighbours, all the same distance from each other. Got that?’
‘Yes.’
‘Now keep on adding dots, extending out in all directions, as far as you can imagine — each dot having six neighbours.’
‘With you so far.’
‘You should have something resembling a Chinese chequerboard. Now concentrate on a single dot again, near the middle. Draw a line from it to one of its six neighbours, and then another line to one of the two dots either side of the neighbour you just chose. Then join the two neighbouring dots. What have you a got?’
‘An equilateral triangle.’
‘Good. That’s three taken care of. Now imagine that the triangle’s sides are twice as long. How many dots are connected together now?’
Childe answered after only a slight hesitation, ‘Six. I think.’
‘Yes.’ Celestine turned to me. ‘Are you following, Richard?’
‘More or less…’ I said, trying to hold the shapes in my head.
‘Then we’ll continue. If we triple the size of the triangle, we link together nine dots along the sides, with an additional dot in the middle. That’s ten. Continue — with a quadruple-sized triangle — and we hit fifteen.’ She paused, giving us time to catch up. ‘There are eight more; up to ninety-one, which has thirteen dots along each side.’
‘The final groove,’ I said, accepting for myself that whatever this problem was, Celestine had definitely understood it.
‘But there are only seven deep grooves in that interval,’ she continued. ‘That means all we have to do is identify the groove on the right which corresponds to the missing triangular number.’
‘All?’ Hirz said.