The likely answer is that this is a seating-design puzzle, not a riddle with a single-object answer. Henry Ernest Dudeney’s puzzle “King Arthur’s Knights” in Amusements in Mathematics is the strongest match: it asks how Arthur and six knights can sit over three evenings without repeating a neighboring pair, while keeping Beleobus near Arthur and Galahad far away. The exact phrase “the treacherous riddle of the round table” does not appear in the transcription, so this identification is likely rather than confirmed.
What the round-table puzzle asks
The transcribed puzzle names seven people: King Arthur and six knights—Beleobus, Caradoc, Driam, Eric, Floll, and Galahad. They meet at the Round Table on three successive evenings. The first evening’s seating is fixed: the knights sit in alphabetical order around the table. On the next two evenings, nobody may sit beside anyone who was already their neighbor on an earlier evening.
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Arthur also has two preferences: he wants Beleobus as near to him as possible and Galahad as far away as possible. The source does not define how to rank schedules if the best possible placement for one conflicts with the best placement for the other.
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At a round table, each person has two neighbors, and the end of the seating order joins back to its beginning. A seating therefore creates a cycle of neighboring pairs, not a straight line. The rule applies to each pair regardless of which person is named first: if Arthur sat beside Beleobus once, that pair cannot be neighbors again.
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The first evening fixes which pairs are already forbidden. Each of the following evenings must avoid all pairs used on every preceding evening—not merely the immediately prior seating. That cumulative restriction is what turns a simple rearrangement into a constrained design problem.
What can—and cannot—be given as the answer
The transcription of Dudeney’s puzzle provides the challenge but not its solution. No verified schedule is established here, so an exact three-evening seating order would be an unsupported answer. Nor does the wording settle what “near” and “far” mean beyond ordinary distance around the circle, or whether “best advantage” prioritizes Beleobus’s placement, Galahad’s placement, or balances both.
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A defensible solution needs to state those conventions, list the circular order for all three evenings, and check every neighboring pair against the earlier evenings. Without that verification, a plausible-looking arrangement is not enough to show the rule has been satisfied.
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The strongest match is puzzle 263, “King Arthur’s Knights,” attributed to Henry Ernest Dudeney in a transcription of Amusements in Mathematics. The transcription says that on the first evening the knights sat in alphabetical order and asks how Arthur could seat them to best advantage under the no-repeat-neighbor rule. The match is not certain because the supplied title wording is not found in that source, and the transcription has not been checked against a primary scan.
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