Vault

Hangman: Solving Both Sides — and the One Parameter That Separates Two Methods

Ballpark Figures solves hangman’s guessing side exactly (backward induction, certified bounds) and then does what a Wordle solver structurally cannot: solves the chooser. Set against 3Blue1Brown’s entropy-based Wordle bot, the two methods turn out to differ in exactly one parameter — how much they trust the word list — and the most robust artifact the exact solver produced is the one built for humans to read.

Sources:

⚠ Sourcing note. The hangman script is a pre-production draft — it carries FILL IN, TALK HERE, and CONFIRM THIS markers. Verbatim prose is trustworthy; blanks are not. Quantities below come from the blog post and the repo’s committed data notes, not from script placeholders.

Links: Battleship — 30 Billion Boards (the parent specimen — same creator; this page inverts half its conclusion), Yahtzee — 259 Trillion → 405 Million (same creator, the state-space-collapse sibling), Symmetry Breaking, The Dominance-Frontier Lens, Monopoly — Frontier Trade Theory (the frontier form of the same collapse-the-search-space thread), Catan — 47k Empirical, The Hollow Opponent

The structural fact that generates everything else

Checked mechanically across both full texts:

Neither author is avoiding the other’s vocabulary. They are describing different objects. Wordle as presented has no second player, so the entire problem is a decision problem against a fixed prior — best-response, no minimax. Hangman has someone sitting across the table, so half the video is a game-theoretic search with no Wordle counterpart.

This is the best-response vs. minimax split reappearing with language as the substrate instead of geometry — and 3b1b’s Wordle video is a pure instance of the guessing half, with no placement half available to it.

But Wordle does have a chooser

3b1b says so in passing: the ~2,315 answers are “a human curated list, I think specifically by the game creator’s girlfriend.” That is a real mixed strategy over words, chosen by a person — optimized for fun, not for difficulty. Wordle isn’t a puzzle by nature; it is a two-player game in which one side has been playing cooperatively and nobody modelled it.

Both authors reach that fork. Ballpark names it — via Jan Misali, that the hangman chooser “isn’t really trying to win as much as they are trying to make the game interesting” — and then says “I am going to completely ignore that fact for the rest of the video.” 3b1b never names it. Same fork, opposite branch, and that single choice generates every other difference below.

Proxy objective vs. true objective

  3b1b / Wordle Ballpark / Hangman
Optimizes expected information (entropy) — a proxy expected misses / win probability — the objective
Method greedy on entropy, 1-ply (2-ply at best) exact backward induction over the decision tree
Endgame needs a patch — entropy will spend the last guess on a word that cannot win falls out as the recursion’s base case
Result 4.124 → 3.6 (with prior) → 3.43 (with answer list + 2-ply) optimal or certified-bounded for every word length

Ballpark identifies why the proxy is nearly free in his game: there are many ways for a letter to be in a word but only one way for it not to be, so the hit-maximizing guess is usually already the information-maximizing guess. That coincidence is why greedy is near-optimal in hangman and merely decent in Wordle. Wordle’s 3⁵-way feedback partition pulls the two apart, which is exactly where 3b1b has to build machinery — a fitted regression f(bits) → expected remaining guesses — to decide when to stop learning and go for the win.

He also settles a standing claim: Jan Misali conjectured greedy was optimal for hangman. Verdict — “wrong in theory, but mostly correct in practice.” The exact solve’s headline gain over greedy is ~6% at length 11 (0.500 → 0.469 misses, exact).

Why hangman is solvable and Wordle isn’t — the size illusion

Hangman looks bigger (34,483 words, lengths 1–20+) and Wordle looks tiny (five letters). Tractability runs the other way, and the reason is branching, not size:

This is the vault’s recurring collapse-the-search-space thread (Yahtzee 259T → 405M; Battleship 30,093,975,536 enumerated; Monopoly’s frontier) — and hangman contributes the sharpest version of its lesson: what determines tractability is the branching factor and the rate at which distinct paths reconverge, not the nominal size of the space. Ballpark expected the easy video precisely because he measured size (six orders of magnitude below Battleship) rather than shape.

The chooser’s counter-move: collapse the space as a weapon

Ballpark hill-climbs a word distribution — multipliers on single-letter blank patterns (BALLPARK = B_______ × _A___A__ × __LL____ × …) — re-solving the guesser each iteration.

The mechanism is an inversion worth naming. Longer words are easier; so the chooser’s counter is to collapse a 7-letter game into a 4-letter game by forcing the tail — trading a large word list for a smaller, harder-conditioned one. Everywhere else in this cluster, state-space reduction is the solver’s tool. Here it is the adversary’s weapon: the same operation, pointed the other way.

The flatness reversal — and what bounds an equilibrium

Battleship’s minimax optimum is ≈ maximum entropy: be uniform or be exploited. Hangman’s is the opposite — a hard concentration on -ING. Why?

You would expect equilibrium to make the guesser indifferent across all 26 letters. It doesn’t: only ~15–16 first letters end up in a virtual tie at length 7, and they are just the already-good letters. Ballpark’s explanation is the payoff of the whole video: “Z just can’t do the things that S, or N, or even P can do.” Inflating Z-words restricts the lexicon so much that the rest of the word becomes easier to guess.

The indifference set of an equilibrium is bounded by the structure of the substrate. Battleship’s grid is homogeneous enough that indifference reaches everywhere — every ship is interchangeable across a uniform surface. English is structurally lumpy: letters have non-substitutable roles, so no re-weighting can equalise them.

This is the natural extension of the symmetry-breaking reading. Battleship showed structure emerging from apparent nothing — a uniform joint distribution inducing a sharply peaked marginal, and non-arbitrary optimal play falling out of it. Hangman is the complementary case: a substrate that is already lumpy, where the lumpiness is load-bearing and cannot be optimized away. Together they bracket the claim: structure creates norms, and the amount of structure sets how far strategic freedom can reach.

(Source-internal inconsistency: the blog says “only 16 letters” in one paragraph and “top 15” in another. Read as ~15–16.)

The one parameter: confidence in the lexicon

The sharpest way to state the difference between the two methods is that they disagree about exactly one thing.

So the entropy bot guessing an impossible word — the thing that looks like a flaw — is correct under uncertainty about the dictionary, and wrong only if you trust it. It isn’t a defect; it’s the premium on the insurance. And in a game where, in Ballpark’s own words, “people can’t even agree on what a word is,” dictionary mismatch is not a corner case — it is the normal condition of a real game between two humans.

Both authors concede the lexicon is the soft underbelly. Ballpark: “a hangman strategy is only as good as its word list” (wordfreq → drop non-letters → hunspell → hand/LLM filtering → 34,483; “phylogeny” makes the cut, “cookout” doesn’t). 3b1b builds his prior from Google Books n-grams pushed through a sigmoid whose parameters he set by “licking my finger and sticking it into the wind.”

The distribution problem, and the escape only hangman offers

Both hit “what distribution does a human actually draw from?” and answer in mirror image. 3b1b models it (frequency + sigmoid cutoff). Ballpark refuses to, keeping uniform because any weighting just helps the guesser and the truth is “somewhere in between.”

The reason he can refuse is structural and elegant: in hangman the word distribution is not a modelling assumption, it is the opponent’s strategy. You don’t estimate it — you solve for it. That option does not exist in Wordle.

The robustness gradient — and why the human-readable artifact wins

Ballpark’s video contains a gradient nobody states outright:

Artifact Depends on Robustness to a wrong word list
Optimal decision tree the exact 34,483-word list + uniform prior brittle — can become undefined
Greedy strategy roughly-right letter frequencies moderate
The alphabet tier list a letter ordering only most robust

And he tested it: re-weighting by frequency shuffles letters within tiers but barely across them; British English moves A past I and drops Z to the bottom (it loses -IZE).

The compression built for human legibility is also the model-error-hardened version of the policy. The tier list is not a lossy concession to video format — it is the artifact that survives contact with a real opponent’s vocabulary.

The tier list is itself derived, not hand-authored: rank letters by a pairwise “is α guessed before β across all words” relation, greedily grow a set minimizing misordered pairs against its complement, and cut a tier wherever one letter wins by a lot. Feed it any solved tree and it regenerates — he says he will reuse it for future word games. Architecture as the product: not an answer, a compression pipeline.

He is also honest that it fails against the optimized chooser — patterns become length-specific with “nothing worth explaining.” The human-portable artifact is valid only against the random chooser. A stated negative result.

Rigor: certified bounds vs. point estimates

3b1b reports simulation point estimates plus a back-of-envelope impossibility argument (start ~11 bits; best two-guess information ~10 bits; ~1 bit remains; so you cannot average 3).

Ballpark reports certified intervals via an α-cap sandwich: cap misses at α and solve → a lower bound; play the capped-optimal policy then fall back to greedy → an upper bound; and if the capped optimum never reaches α, the bound is tight and optimality is proved. Lengths 3–6 stayed open — and those points are plotted in red, so the chart itself distinguishes solved from bounded.

The visualization carries the uncertainty. That is a transferable standard, not a hangman detail.

Rules design falls out of solving both sides

Because both sides are solved, he can compute what a Wordle video cannot: what the rules should be. Under near-optimal play the fair miss allowance runs 1–8 depending on word length — every number except 6, which is the one everybody plays. The standard game is miscalibrated, and the correction is length-dependent.

Three thresholds worth keeping straight:

Open Questions

Tags

games · game-theory · game-design · mathematics · strategy