I tried the chemosensor against part of my own suit.
It had stopped working.
“Fix that,” I instructed my suit, authorising it to divert whatever resources it required to the task.
“Problem, Richard?” asked Childe.
“My suit’s damaged. Minor, but annoying. I don’t think the Spire was too thrilled about my taking a sample of it.”
“Shit. I probably should have warned you of that. Argyle’s lot had the same problem. It doesn’t like being cut into, either. I suspect you got off with a polite warning.”
“Generous of it,” I said.
“Be careful, all right?” Childe then told everyone else to disable their chemosensors until told otherwise. Hirz grumbled, but everyone else quietly accepted what had to be done.
In the meantime I continued my own survey of the room, counting myself lucky that my suit had not provoked a stronger reaction. The chamber’s circular wall was fashioned from what looked like the same hard, dull alloy, devoid of detail except for the point where it framed what was obviously a door, raised a metre above the floor. Three blocky steps led up to it.
The door itself was one metre wide and perhaps twice that in height.
“Hey,” Hirz said. “Feel this.”
She was kneeling down, pressing a palm against the floor.
“Careful,” I said. “I just did that and—”
“I’ve turned off my chemo-whatsit, don’t worry.”
“Then what are you—”
“Why don’t you reach down and see for yourself?”
Slowly, we all knelt down and touched the floor. When I had felt it before it had been as cold and dead as the floor of a crypt, yet that was no longer the case. Now it was vibrating; as if somewhere not too far from here a mighty engine was shaking itself to pieces: a turbine on the point of breaking loose from its shackles. The vibration rose and fell in throbbing waves. Once every thirty seconds or so it reached a kind of crescendo, like a great slow inhalation.
“It’s alive,” Hirz said.
“It wasn’t like that just now.”
“I know.” Hirz turned and looked at me. “The fucking thing just woke up, that’s why. It knows we’re here.”
THREE
I moved to the door and studied it properly for the first time.
Its proportions were reassuringly normal, requiring only that we stoop down slightly to step through. But for now the door was sealed by a smooth sheet of metal, which would presumably slide across once we had determined how to open it. The only guidance came from the door’s thick metal frame, which was inscribed with faint geometric markings.
I had not noticed them before.
The markings were on either side of the door, on the uprights of the frame. Beginning from the bottom on the left-hand side, there was a dot—it was too neatly circular to be accidental—a flat-topped equilateral triangle, a pentagon and then a heptagonal figure. On the right-hand side there were three more figures with eleven, thirteen and twenty sides respectively.
“Well?” Hirz was looking over my shoulder. “Any bright ideas?”
“Prime numbers,” I said. “At least, that’s the simplest explanation I can think of. The number of vertices of the shapes on the left-hand frame are the first four primes: one, three, five and seven.”
“And on the other frame?”
Childe answered for me. “The eleven-sided figure is the next one in the sequence. Thirteen’s one prime too high, and twenty isn’t a prime at all.”
“So you’re saying if we choose eleven, we win?” Hirz reached out her hand, ready to push her hand against the lowest figure on the right, which she could reach without ascending the three steps. “I hope the rest of the tests are this simp—”
“Steady, old girl.” Childe had caught her wrist. “Mustn’t be too hasty. We shouldn’t do anything until we’ve arrived at a consensus. Agreed?”
Hirz pulled back her hand. “Agreed . . .”
It took only a few minutes for everyone to agree that the eleven-sided figure was the obvious choice. Celestine did not immediately accede; she looked long and hard at the right-hand frame before concurring with the original choice.
“I just want to be careful, that’s all,” she said. “We can’t assume anything. They might think from right to left, so that the figures on the right form the sequence which those on the left are supposed to complete. Or they might think diagonally, or something even less obvious.”
Childe nodded. “And the obvious choice might not always be the right one. There might be a deeper sequence—something more elegant—which we’re just not seeing. That’s why I wanted Celestine along. If anyone’ll pick out those subtleties, it’s her.”
She turned to him. “Just don’t put too much faith in whatever gifts the Jugglers might have given me, Childe.”
“I won’t. Unless I have to.” Then he turned to the infiltration specialist, still standing by the frame. “Hirz—you may go ahead.”
She reached out and touched the frame, covering the eleven-sided figure with her palm.
After a heart-stopping pause there was a clunk, and I felt the floor vibrate even more strongly than it had before. Ponderously, the door slid aside, revealing another dark chamber.
We all looked around, assessing each other.
Nothing had changed; none of us had suffered any sudden, violent injuries.
“Forqueray?” Childe said.
The Ultra knew what he meant. He tossed the float-cam through the open doorway and waited several seconds until it flew back into his grasp.
“Another metallic chamber, considerably smaller than this one. The floor is level with the door, so we’ll have gained a metre or so in height. There’s another raised door on the opposite side, again with markings. Other than that, I don’t see anything except bare metal.”
“What about the other side of this door?” Childe said. “Are there markings on it as well?”
“Nothing that the drone could make out.”
“Then let me be the guinea pig. I’ll step through and we’ll see what happens. I’m assuming that even if the door seals behind me, I’ll still be able to open it. Argyle said the Spire didn’t prevent anyone from leaving provided they hadn’t attempted to access a new room.”
“Try it and see,” Hirz said. “We’ll wait on this side. If the door shuts on you, we’ll give you a minute and then we’ll open it ourselves.”
Childe walked up the three steps and across the threshold. He paused, looked around and then turned back to face us, looking down on us now.
Nothing had happened.
“Looks like the door stays open for now. Who wants to join me?”
“Wait,” I said. “Before we all cross over, shouldn’t we take a look at the problem? We don’t want to be trapped in there if it’s something we can’t solve.”
Childe walked over to the far door. “Good thinking. Forqueray, pipe my visual field through to the rest of the team, will you?”
“Done.”
We saw what Childe was seeing, his gaze tracking along the doorframe. The markings looked much like those we had just solved, except that the symbols were different. Four unfamiliar shapes were inscribed on the left side of the door, spaced vertically. Each of the shapes was composed of four rectangular elements of differing sizes, butted together in varying configurations. Childe then looked at the other side of the door. There were four more shapes on the right, superficially similar to those we had already seen.
“Definitely not a geometric progression,” Childe said.
“No. Looks more like a test of conservation of symmetry through different translations,” Celestine said, her voice barely a murmur. “The lowest three shapes on the left have just been rotated through an integer number of right angles, giving their corresponding forms on the right. But the top two shapes aren’t rotationally symmetric. They’re mirror images, plus a rotation.”
“So we press the top right shape, right?”
“Could be. But the left one’s just as valid.”
Hirz said, “Yeah. But only if we ignore what the last test taught us. Whoever the suckers were that made this thing, they think from left to right.”
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 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.
“Look, it’s simple. I know the answer, but you don’t have to take my word for it. The triangles follow a simple sequence. If there are N dots in the lower row of the last triangle, the next one will have N plus one more. Add one to two and you’ve got three. Add one to two to three, and you’ve got six. One to two to three to four, and you’ve got ten. Then fifteen, then twenty-one . . .” Celestine paused. “Look, it’s senseless taking my word for it. Graph up a chequerboard display on your suits—Forqueray, can you oblige?—and start arranging dots in triangular patterns.”
We did. It took quarter of an hour, but after that time we had all—Hirz included—convinced ourselves by brute force that Celestine was right. The only missing pattern was for the fifty-five-dot case, which happened to coincide with one of the deep grooves on the right side of the door.
It was obvious, then. That was the one to press.
“I don’t like it,” Hirz said. “I see it now . . . but I didn’t see it until it was pointed out to me. What if there’s another pattern none of us are seeing?”
Celestine looked at her coldly. “There isn’t.”
“Look, there’s no point arguing,” Childe said. “Celestine saw it first, but we always knew she would. Don’t feel bad about it, Hirz. You’re not here for your mathematical prowess. Nor’s Trintignant, nor’s Forqueray.”
“Yeah, well remind me when I can do something useful,” Hirz said.
Then she pushed forward and pressed the groove on the right side of the door.
Progress was smooth and steady for the next five chambers. The problems to be solved grew harder, but after consultation the solution was never so esoteric that we could not all agree on it. As the complexity of the tasks increased, so did the area taken up by the frames, but other than that there was no change in the basic nature of the challenges. We were never forced to proceed more quickly than we chose, and the Spire always provided a clear route back to the exit every time a doorway had been traversed. The door immediately behind us would seal only once we had all entered the room where the current problem lay, which meant that we were able to assess any given problem before committing ourselves to its solution. To convince ourselves that we were indeed able to leave, we had Hirz go back the way we had come in. She was able to return to the first room unimpeded—the rear-facing doors opened and closed in sequence to allow her to pass—and then make her way back to the rest of us by using the entry codes we had already discovered.