While Richard Swift is visiting the Monument to the Eighty in Chasm City, where he meets his old friend Roland Childe, who has been presumed dead for over a century and a half. Childe takes Richard back to his home, and reveals that he is assembling a team to tackle a curious artificial - alien - structure found by probes sent out secretly by his family ages ago.
The team consisted of Richard, Celestine (Richard's ex-wife who underwent Pattern Juggler neural transforms that left her with a brilliant capacity at mathematics, and who divorced Richard around 2490), Hirz (sometime hacker, sometime infiltrator, who has herself frozen between missions), Dr. Trintignant (expert doctor and cyberneticist, infamous for conducting horrific medical experiments on allegedly unconsenting subjects), Forqueray (an Ultranaut, captain of the lighthugger Apollyon) and, of course, Childe himself.
From Chasm City, they travel to the alien artifact, which they promptly name "Blood Spire". Remains of the previous human explorers to visit the place lie around - supposedly they belonged to another Ultranaut crew led by a captain called Argyle, whom Childe's probe interrogated during his dying moments to gather information about the Spire.
The Spire is a series of rooms, each containing a mathematical puzzle. The doors get smaller as the rooms progress, and the rooms proceed in a spiral up the tower, which is about 250 meters high. The tower, interestingly, floats off the surface of the planet without any detectable force or support holding it up. The puzzles cover most of mathematics, with various questions tackling triangular numbers, rotations of four-dimensional figures and their corresponding shadows, and arcane aspects of prime number theory. It is not known what the Spire guards, or why there should be so many puzzles.
Disturbingly, the Spire also seems to be alive. Initially cold and silent, it "woke up" and started to warm up and vibrate once Childe's crew entered the structure. It also inflicts painful and often gruesome punishments for getting wrong answers or going over some unspecified time limit (which becomes shorter with each puzzle). Forqueray and Hirz are killed by these punishments.
To deal with the Spire's puzzles, the team submit to more and more cybernetic and artificial aids from Trintignant, which eventually culminate with Childe and Richard resembling nothing so much as diamond dogs, with artificially-accelerated consciousness and an advanced grasp of mathematics. Celestine abandoned the quest earlier.
While tackling the Spire, Celestine barges in and tries to persuade Richard to abandon the quest. Apparently, Childe knew more about the Spire than he should, and medical investigation of the corpses revealed all of them came from the same individual - because they had the same DNA. It's revealed that the bodies were actually clones of Childe, who had already visited the place before, and what he did was to go in, get to where he thought he could not go on much further, and then have his memories trawled and implanted into a clone, before returning to continue solving the puzzles, and die of failure. Counting the original, the Childe then in the Spire with Richard and Celestine was the nineteenth in the series.
Finally, Richard abandons the quest only to find that Dr. Trintignant, confronted with the possibility of restoring Richard to his human form and thus undoing his magnum opus, decided to commit suicide by disassembly. There was surprisingly little organic matter left among his remains, which were all sorted and placed neatly in jars and the like. Richard and Celestine end up going back to post-plague Yellowstone - when they left they could not find any new remains of Childe, who was thus presumably still inside the Spire - and Richard is left without any hope of becoming human again, since the Melding Plague wiped out most of the required technology.
In the end, Richard, faced with the sheer temptation and curiosity of the Spire, secretly slips away and hires the lighthugger Poseidon to take him "somewhere".