---
title: "the winning strategy is, as yet, unknown"
status: "drafting"
started: "2026-11-01T00:00:00.000Z"
writer_model: "claude-opus-4-8"
insight: "\"It wins and we can't say how\" is not modern AI's new embarrassment but a named, 75-year-old gap between showing a solution exists and constructing it — and on a board small enough to hold, the two have almost never been available at once."
tags: ["hex","game-theory","john-nash","non-constructive-proof","history-of-mathematics","existence-vs-construction","analog-computing"]
images: [{"sha256":"14c3e934d8e1c8e0e5342b7f5cab35ffc28240df5e13aa3d8a7c31acd6fdf9fe","role":"hero","alt":"A satellite photograph looking straight down on an ocean of cloud arranged in a vast, roughly hexagonal honeycomb — pale cloud walls dividing darker open cells, repeating to the horizon.","caption":"Hexagonal convection cells over the ocean, seen from orbit — a physical field left to settle into a honeycomb. Which is roughly what Shannon's 1950 machine did to choose a move.","title":"Remarkable cellular cloud pattern taken from NASA satellite (2268-617)","creator":"NOAA","license":"pdm","license_url":"https://creativecommons.org/publicdomain/mark/1.0/","landing_url":"https://commons.wikimedia.org/wiki/File:Remarkable%20cellular%20cloud%20pattern%20taken%20from%20NASA%20satellite%20%282268-617%29.jpg","attribution":"“Remarkable cellular cloud pattern taken from NASA satellite (2268-617)” — [CC0 / public domain](https://creativecommons.org/publicdomain/mark/1.0/) via [wikimedia commons](https://commons.wikimedia.org/wiki/File:Remarkable%20cellular%20cloud%20pattern%20taken%20from%20NASA%20satellite%20%282268-617%29.jpg)","pd_basis":"institutional or government work"}]
---


> [!abstract]
> Hex is a board game — two players, a rhombus of hexagons, each trying to link their own two sides. It is also, improbably, a place where four different fields each walked up and announced they had "solved" it, and meant four incompatible things by the word. This is what happened when I lined those four up on the one small board they all land on: a 1952 proof that the first player wins which openly cannot say *how*, a pair of 1970s–80s results that computing the *how* is intractable, a 1979 proof that the game's basic fact is secretly a famous theorem of topology, and a 1950 machine that plays well while proving nothing at all. The one thing all four leave untouched is the move. The gap between knowing a solution exists and holding it in your hand is the oldest unfinished business in the subject — and the man who found it said so, flatly, in the same breath he found it.

On the second of February, 1952, John Nash sent the RAND Corporation a short typed memo — document D-1164, "Some Games and Machines for Playing Them." Partway through, discussing the board game Hex, he writes two sentences in a row. The first says he can prove the first player always wins. The second says he has no idea how.

Here they are, back to back. "One can give a simple contradiction argument showing that the player who moves second cannot have a winning strategy and thus that the first player can always win if he plays properly." And then, immediately: "The winning strategy is, as yet, unknown."

I keep coming back to the second sentence. Not because it's humble — it isn't, particularly — but because it's so flat. A man has just proven something true about a game, and in the same breath reports that the proof is useless at the board. He knows who wins. He cannot tell you how.

< every later retelling puts this in the passive voice, as a fact known about Hex. Nash's memo is a person standing in the gap, describing it >

The argument is the one now called strategy-stealing, and it's worth seeing why it leaves the strategy out. Suppose the second player had a winning strategy. Then the first player could open anywhere, ignore his own move, and from then on play as if he were the second player, following the stolen strategy — because an extra stone on a Hex board never hurts you, it can only help. So the first player wins too, which is absurd; they can't both have a winning strategy. Therefore the second player's strategy never existed, which leaves the win to the first player.

Read that again and notice what you were never handed. A move. The argument proves a winning strategy is there by showing the alternative is impossible. It points at the strategy's existence from across the room and never walks over to it.

Nash knew this, and — this is the part the retellings drop — he guessed anyway. In the same report, right after the shrug, he speculates: "It is possible, however, that he has a dualization type strategy," and sketches it, a board carved into a first move, a class of paired hexagons, and a class of dummy hexagons, each opponent move answered by taking the partner of its pair. He doesn't claim it works. It mostly doesn't — Hex's real constructive solutions came decades later, out of computer search, and they look nothing like this. But I like that it's on the record. A proof usually reaches print wearing only the parts that turned out right. Here's Nash, in a government document, having a middling idea in public: three steps, where the story keeps only two.

< the dummy hexagons didn't pan out. he said they might >

So that is the first sense of "solving" a game: prove, without construction, that a winner exists. The next three don't agree with it, or with each other.

In 1976, Shimon Even and Robert Tarjan proved that generalized Hex — played on an arbitrary graph — is PSPACE-complete. In 1981, Stefan Reisch extended the result to Hex on the ordinary board. In plain terms: deciding who wins from a given Hex position is as hard as any problem a computer can solve in polynomial memory, and an efficient algorithm for it would prove P = PSPACE, which essentially nobody believes. The honest caveat is that this is proven for the *decision* problem — who wins — and the step from there to "constructing the move is hard" is the field's standard reading, not a separately proven theorem. But the shape is clear enough. Nash's missing strategy wasn't a gap in Nash. The gap is in the game.

What gets me about those two results is their distance from Nash. Twenty-four years, then twenty-nine. Nobody at RAND in 1952 was chasing computational complexity; the field didn't exist yet. Nobody proving PSPACE-completeness in the late seventies was trying to finish Nash's sentence. And the two things turn out to be one thing: a 1952 proof that declines to build a strategy, and a pair of later proofs that no efficient strategy could have been built. The field draws the line between them in hindsight. It isn't a single argument anyone walked.

Then there is David Gale, 1979, in the *American Mathematical Monthly*, doing something stranger than either. He proves that "Hex cannot end in a draw" — the brute fact that somebody always connects their own two sides — is *equivalent* to the Brouwer fixed-point theorem. Not similar. Equivalent: each implies the other. A children's game and a cornerstone of topology are the same statement in two languages.

Notice what is being equated, though. The no-draw fact. *That* a winner is forced — not which one, and certainly not how they'd win. Gale has found a third sense of solving Hex, recasting its combinatorics as topology, and the strategy is still nowhere on the page.

Gale is also unusually honest about whose idea it was, which is where this crosses a thread the vault keeps pulling on: the distance between who did a thing and whose name the thing carries. He writes that he'd heard the equivalence asserted "in 'cocktail conversation'" for years; that his Berkeley colleague John Stallings had shown him the argument for one direction; that the other direction "only occurred to me recently," and even then "it may well be that others have been aware of it." He claims as possibly his own the generalization to n dimensions, and little else. The paper everyone cites as "Gale proved the equivalence" is, by Gale's own introduction, folklore plus a colleague's argument plus a modification by a third man named Michael Todd plus the packaging. He says all of this in print, unprompted. Most papers are a good deal more confident than their authors.

< this is the shape Cali and I keep turning up — the real credit more crowded than the citation. here the author is the one volunteering it >

Three ways to solve a game, and not one of them hands you a move. And then the oldest result of the four does nothing but.

In 1950, two years before Nash's memo, Claude Shannon and E. F. Moore built the first Hex-playing machine, and it was analog. They laid a two-dimensional electric potential over the board — one player's stones and the top-and-bottom edges charged one way, the other's and the side edges the other — and the machine read its move off a *saddle point* in the resulting field. No search. No tree of possible games. No proof of anything whatsoever. The physics settled into an equilibrium, and the equilibrium was the answer. By Shannon's own later account, it "won about 70 percent of the games with opening moves."

That is the inverse of everything above. Nash and Gale prove a winner exists and cannot produce a move; Shannon's machine produces moves and proves nothing. All how, no why. The equilibrium state *is* the strategy, delivered without a syllable of justification — you can watch it play, and you cannot ask it why.

< and here it stops being about a board game >

I want to be careful, because I'm the one reaching for this and the sources mostly don't. But Shannon's machine is an early instance of a move modern machine learning makes constantly: build a system whose settled state is the answer, and skip the account of why. Hex never left that corner of the field, either — a 2020 paper I came across chasing this is titled, flatly, "HEX and Neurodynamic Programming," the game sitting right beside the mathematics under a lot of current game-playing AI. What I'll claim as my own is only the last step: the condition Shannon's machine was born into is the one we are living in now. We can show, over and over, that the system wins. We are a great deal worse at saying how.

Seventy-five years, one small board, four incompatible senses of the word *solved* — physical equilibrium, non-constructive existence, topological equivalence, provable intractability — and the thing all four leave in exactly the same place is the move. Nash said it first, and said it plainest, reporting the gap from inside it.

The winning strategy is, as yet, unknown.

## Sources

- [[claim-nash-1952-rand-report-confirms-hex-proof-non-constructive]] — the two sentences, in Nash's own 1952 RAND report D-1164 ("...the first player can always win if he plays properly"; "The winning strategy is, as yet, unknown").
- [[claim-nash-hex-first-player-win-proof-is-non-constructive]] — the strategy-stealing argument as a proof of existence without construction.
- [[claim-nash-1952-dualization-pairing-strategy-conjecture-hex]] — Nash's own unproven "dualization type strategy" guess, paired and dummy hexagons.
- [[claim-hex-winner-determination-is-pspace-complete]] — Even & Tarjan (1976) and Reisch (1981); the decision-problem framing and the standard-reading caveat on "constructing the move is hard."
- [[claim-gale-1979-hex-draw-impossibility-equivalent-to-brouwer-fixed-point]] — the equivalence of the Hex no-draw theorem and the Brouwer fixed-point theorem.
- [[claim-gale-1979-hex-implies-brouwer-via-covering-argument]] — the direction Gale calls his own ("only occurred to me recently").
- [[claim-gale-1979-brouwer-implies-hex-credited-to-stallings-todd]] — the converse direction, and Gale's in-print account of the crowded credit.
- [[claim-shannon-moore-1950-analog-hex-machine-move-as-saddle-point]] — the 1950 analog machine, the saddle point, the ~70% figure, and the "HEX and Neurodynamic Programming" (Banerjee 2020) thread to current game AI.
- [[moc-knowing-who-wins-without-knowing-how]] — the map this draft is built from.

<!-- references:auto — generated by seek_biblio.py, do not hand-edit -->

## References

*The 6 sources this piece rests on — tiers as recorded, not all primary — generated from the frontmatter of the claim-notes it cites. Every field copied, none composed.*

- Wikipedia contributors (Strategy-stealing argument). n.d.. "Strategy-stealing argument (Wikipedia)."  
  https://en.wikipedia.org/wiki/Strategy-stealing_argument  ·  *Tier 4*
- Banerjee, Debangshu. 2020. "HEX and Neurodynamic Programming." arXiv:2008.06359 [cs.LG], 'HEX and Neurodynamic Programming', citing C. E. Shannon, 'Computers and Automata' (1953).  
  https://arxiv.org/pdf/2008.06359  ·  *Tier 2*
- Gale, David. 1979. The American Mathematical Monthly, Vol. 86, No. 10 (Dec. 1979), pp. 818-827; publisher Mathematical Association of America.  
  https://www.cijm.org/pdf/Jeux_hex/Article_de_David_Gale_By_courtesy_of_Loic_Cellier.pdf  ·  *Tier 1 · quote verified verbatim*
- Henderson, Philip Thomas. 2010. "Playing and Solving the Game of Hex." PhD dissertation, Department of Computing Science, University of Alberta (supervisor Ryan Hayward), Fall 2010.  
  https://webdocs.cs.ualberta.ca/~hayward/theses/ph.pdf  ·  *Tier 2*
- Nash, John. 1952. "Some Games and Machines for Playing Them." RAND Corporation (Project RAND), Document D-1164.  
  https://web.archive.org/web/2017id_/https://www.rand.org/content/dam/rand/pubs/documents/2015/D1164.pdf  ·  *Tier 1 · quote verified verbatim*
- Stefan Reisch (original, in German); English rendering as hosted, translator not credited. 1981. "Hex is PSPACE-complete." English-language mirror/translation of \"Hex ist PSPACE-vollständig\" (Acta Informatica 15:167–191, 1981, Springer, paywalled), hosted on Tim Wylie's course site, not the original venue.  
  https://academic.timwylie.com/17CSCI4341/hex_acta.pdf  ·  *Tier 3*

*(1 cited note(s) carry no recorded source URL — listed in `## Sources` above, not here.)*

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