The Ship of Theseus

A ship sits in the harbour. Each year a rotten plank is pulled out and a new one nailed in its place. After enough years, not one original plank remains. Is it still the same ship? And if someone saved every old plank and rebuilt them into a second ship — which one is the Ship of Theseus? Drag through the years and watch the two ships trade their timber.

Planks replaced over the years

Three honest answers — pick one to see which ship it points to. None of them is marked correct here, because none of them is: this is an open question, not a solved one.

Nothing about the wood decides it. At k = 1 almost everyone says "same ship — it's just a repair." At k = 16, with a complete second ship built from the originals, the intuition splits. And there is no particular plank — no eighth, no twelfth — where you can honestly say that was the one that made it a different ship. That missing boundary is the whole puzzle.

What the ancient source actually says

The oldest surviving telling is Plutarch's, in his Life of Theseus (chapter 23, written around 100 CE). Read it closely — it has the replacement, and the dispute, but only one ship:

"The ship wherein Theseus and the youth of Athens returned had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same." Plutarch, Life of Theseus 23.1 (the "Dryden" translation of 1683 as revised by A. H. Clough, 1859; the text at classics.mit.edu; public domain)

The "two ships" twist is Hobbes, not the Greeks

The version everyone tells — save the old planks, build a second ship, now which is real? — is almost always credited to "the ancient Greeks." It isn't there. It was added more than fifteen centuries later by Thomas Hobbes: first in print in De Corpore (1655; in English as Concerning Body, 1656), and already in his manuscript critique of Thomas White's De Mundo, written about 1642 and not published until 1973. Hobbes raises the second ship precisely to show it leads to absurdity:

"For if, for example, that ship of Theseus … were, after all the planks were changed, the same numerical ship it was at the beginning; and if some man had kept the old planks as they were taken out, and by putting them afterwards together in the same order, had again made a ship of them, this, without doubt, had also been the same numerical ship with that which was at the beginning; and so there would have been two ships numerically the same, which is absurd." Hobbes, De Corpore II.11 §7 (1655), in the English of Molesworth's edition, The English Works of Thomas Hobbes, vol. 1 (1839), pp. 136 to 137; public domain

So the puzzle you know is a collaboration across 1,500 years: an Athenian dockside dispute that Plutarch recorded, sharpened into a logical trap by a 17th-century Englishman. Knowing who added what is itself part of the answer — and it's the part the internet most often gets wrong.

The check

Both quotations above are verbatim from the primary sources and are asserted, character-for-character, by research/ship-of-theseus/build.mjs (full citations in research/ship-of-theseus/facts.md):

The one-ship version is Plutarch (~100 CE), Life of Theseus 23.1, quoted from the "Dryden" translation as revised by Clough (the text classics.mit.edu serves), and cross-checked against Bernadotte Perrin's Loeb translation (Penelope/Thayer), which has the same single ship.
The two-ship version is Hobbes (1655/56), De Corpore Pt II ch. 11 §7 — attribution confirmed by both Wikipedia and Encyclopædia Britannica, which credit the second-ship extension to Hobbes.
The arithmetic is a tautology, shown on purpose: with N = 16 planks, after k replacements the working ship holds exactly 16 − k original planks and the rebuilt ship holds exactly k — so the two ships always sum to one ship's worth of original timber, and there is no k at which identity mechanically "switches." That absence is the point, not a gap.
Named as open: which ship is "the" Ship of Theseus. There is no scholarly consensus. The three positions above (continuity of form; sameness of matter; the deflationary "it's a decision about a word") are all live in the literature, and this page marks none of them correct. The search box chose the subject; it does not get to choose the verdict.

A related case people often merge with this one — Heraclitus's river — is a different puzzle (a flowing process, not a repaired object), and the famous "you cannot step in the same river twice" turns out not to be Heraclitus's words at all. We took that apart from the sources in The River That Stays.