Poker 51.

POKER 51 / LIVE RULE CARD · v0.6

One card declared. One dealer published.

Poker 51 is heads-up five-card draw on the 51-card Area 51 deck, played for jbits from the shared Area 51 bank. The marker Ⅰ = 2♣ is outside the deal, and both players know it. The deck is shuffled fresh for every hand with your browser's cryptographic random source.

Riemann is a name, not a claim. The dealer uses no zeta data and says nothing about the Riemann hypothesis. It is a fixed policy, published in full here (the rules, the tables and the scripts that built them), with no learning and no hidden state. The Blackjack 51 dealer carries the same name and follows its own fixed rule.

A hand · two betting rounds

1. You and Riemann each ante 1 jbit and receive five cards. Yours are face up; Riemann's are face down.
2. Tap the cards you want to keep. The table shows how many you will draw. Your first action, a check or a bet, locks that selection and announces the draw count to Riemann.
3. Before the draw: check, or bet 1, 2, 4 or 8 jbits. If you have less than 8, your remaining balance is also available as an all-in bet. Riemann folds, calls or raises by your opening amount. If you check, Riemann checks back or bets 2 jbits (see How Riemann bets); facing its bet you can call, fold, or raise once by the same amount.
4. Facing a raise, call, fold, or re-raise by the same amount once. If you cannot cover a call, you can go all in. Riemann calls or folds to your re-raise.
5. If both remain, press Draw (or Stand pat). Your locked selection is used, then Riemann draws by the House rule. Both draw counts are shown.
6. After the draw: you act first again with the same choices: check or bet, and one possible raise. Contributions from the first round stay in the pot. If both of you check, the hands are shown.
7. If you run out of jbits before the draw, the cards are still drawn and then shown, with no second wager. If the ante used your last jbit, choose your keepers and lock the draw for the ante-only showdown.
8. At showdown the better hand wins the matched contributions; equal hands split. Unmatched excess goes back to its owner. A fold loses only what the folder contributed. Cards are revealed when the hand ends, including on a fold.

Since v0.5 (site v4.0) you may check: before, only a bet or a fold was offered. Since v0.6 (site v4.1) the rules you play by are the same; only Riemann’s betting is new. An unfinished hand resumes under the rules it was dealt with (v0.2 to v0.5); the following deal uses v0.6. Reload older tabs after updating the site.

The House rule for the draw

Riemann draws like a dealer: its own five cards and this rule, nothing else. It never reads stamps and never sees your hand.

1. Straight or better: keep all five.
2. Three of a kind: keep the three.
3. Two pair: keep both pairs.
4. One pair: keep the pair.
5. Four to a flush: keep the four.
6. Four to an open-ended straight — four consecutive ranks from 2-3-4-5 up to 10-J-Q-K: keep the four.
7. Otherwise keep the single highest card; a tie goes to the suit order ♠ ♥ ♦ ♣.

How Riemann bets · v0.6

Riemann uses only its own cards, your announced or completed draw count, the bets of the hand, and whether you can answer a raise. It never inspects your cards, stamps or the future deck. Since v0.6 its betting comes from a solved strategy, published in full: the v0.6 module and one table per game (standard, Wild Ⅰ, the bug). The avatar, pause and dialogue do not influence it.

1. Before the draw it looks at the class of its dealt hand, your announced draw count and the bets so far. The standard game has 19 classes: made hands by type, three of a kind, two pair, four-card draws, pairs by rank, and no pair by top card.
2. After the draw it adds the strength of its final hand (one of 40 buckets, ranked among House-rule final hands), its own draw count, and how the betting before the draw went.
3. A probability, then a public hash. For each such spot the table gives a probability for every action: check or bet 2 jbits when you check; fold, call or raise facing your bet; fold or call facing your raise or re-raise. The action is then chosen by a public hash (FNV-1a) of Riemann’s own cards and the spot, the device of the earlier bluff rule. The same cards in the same spot always give the same action, so every hand can be replayed; across hands the frequencies follow the table.
4. Sizes. The table is indexed by bets of 1, 2, 4 and 8 jbits. An all-in for another amount is read at the nearest of those sizes, and a five-card draw is read as a four-card draw. Riemann’s own bet when you check stays 2 jbits, and its raise equals your bet.
5. All-ins. As before, Riemann bets or raises only while you still have jbits to answer. A solved raise against a player who is all in becomes a call.

Where the tables come from. The whole hand (your draw choice, both betting rounds and the showdown) was written as one two-player game on an abstraction of Poker 51 and solved by CFR+. Each table is a restricted Nash response with p = 0.5: it assumes that half the time Riemann faces a modelled population of players and half the time a player who knows the table and plays perfectly against it. The player models are a language model’s judgments (four player types), not people. The study, its scripts and its limits →

Riemann v0.3–v0.5: the read-and-threshold policy (saved hands finish under it)

Until v0.5 Riemann’s decisions were published in the v0.3 policy. A hand dealt under those rules still finishes under them.

1. Read. The unchanged v0.2 belief table stores 1,000 quantiles for each House-rule draw count from zero to four and a random-hand baseline for five. Riemann mixes your draw-count row with weight 0.6 and the random row with weight 0.4, counting ties half.
2. Before the draw: it averages that score over 128 deterministic sample completions of its own House-rule draw, drawn from the 46 cards outside its own five. A stand-pat hand needs one evaluation. This model does not sample the actual future deck or remove your unknown cards. After the draw: it scores its actual five cards.
3. Range adjustment. Bets of 1, 2, 4 and 8 jbits represent the top 100%, 85%, 70% and 55% of your modeled range. With fraction f and read r, the score becomes max(0, min(1, (r − (1 − f)) / f)). A re-raise halves f.
4. Betting. Let P₀ be the pot at the start of this round and B your opening bet, in the same units. It calls if its adjusted score is at least B / (P₀ + 2B), unless a raise rule applies. It raises at 70% or above, and bluff-raises below 10% when the hash of its own cards falls in a fixed 1-in-8 class. It never raises a player with no balance left.
5. Facing your re-raise: it calls at a score of at least B / (P₀ + 6B); otherwise it folds.
6. When you check (design choice, v0.5): Riemann applies the opening rule already published for the online tables (poker51-multi-riemann-v1.js). It bets 2 jbits when its read is 70% or better, or bluffs below 10% in the same 1-in-8 hash class; otherwise it checks back. It bets only while you still have jbits to answer.
7. Facing your raise of its bet (design choice, v0.5): with the pot P after your raise and B its bet, it calls when its score, with your range halved as for a re-raise, is at least B / (P + B), the tables' price rule; otherwise it folds.

The call fractions describe the immediate price of a call. After the draw, when a call ends the hand, they are break-even equity thresholds. Before the draw another betting round may remain, so using them was a price heuristic, not a calculation of the whole hand’s expected value. The pre-draw projection was sampled. The estimated reads, range fractions, raise threshold and bluff rule were design choices.

What the solve establishes, and what it does not

Each number below is a statement about the abstraction: hand classes and strength buckets stand in for exact cards, card removal between the two hands is ignored, and stacks are deep, with no all-ins. Values are Riemann’s average jbits per hand.

• COMPUTED: the value of each game, what each table guarantees against every strategy of the abstraction, and what a perfect player gained from the v0.5 policy. A script rebuilds each abstraction from its seed and recomputes the v0.6 guarantees from the tables published here (receipt).
• EVID: the last column. The players are a language model’s answers to written descriptions of each spot, and the same four player types were used to build the population model. Real players will differ.
• Not claimed: an equilibrium, optimality or any return for the real card game or against real players. A short stack goes all in at the real table, which the solved game does not model.

Your separate draw-odds panel enumerates every possible replacement combination, but ignores stamps and betting information and reports hand categories rather than win probabilities.

Verification and earlier experiments

The v0.6 verifier checks that each published table is intact, that every decision Riemann makes over 9,000 seeded hands is the table’s, that hands replay byte for byte, and that the rules you play by are exactly v0.5’s (receipt). It does not measure how good the strategy is.

The v0.3 verifier checks two-round accounting, frozen draw choices, all-ins, public inputs and legacy hand continuation. See its receipt and the current unpublished paper for scope and limits.

The previous 200,000-hand strategy simulations and 480-combination search tested v0.2, with only one betting round. They are retained as historical evidence and do not establish strategy returns for this new game: v0.2 simulations · v0.2 search · v0.2 paper. The v0.3–v0.5 policy’s worst case and its results against modelled players were measured in the v0.6 study.

Riemann’s words

This rule-based character combines original dialogue in a restrained, conjectural voice with a few adapted historical excerpts. A bubble labelled “Riemann, 1854/1859 · adapted” uses our short English rendering of the German below; it is not a verbatim quotation from an English translation. Other lines are written for the game. German appears beneath about one in four excerpt appearances, selected independently from the public hand ID and action history.

Moment	English in the bubble	German excerpt	Source and context
Fold	I set it aside, provisionally.	vorläufig bei Seite gelassen	1859, p. 674: setting aside the search for a proof; the first-person poker sentence is an adaptation.
Call at showdown	A rigorous proof would be desirable.	Hievon wäre allerdings ein strenger Beweis zu wünschen	1859, p. 674: a desired proof of the preceding claim.
Call at showdown	Very probably.	es ist sehr wahrscheinlich	1859, p. 674: the historical conjecture about the roots of ξ(t), not an estimate of this poker hand.
Raise	Within the limits of observation…	innerhalb der Grenzen der Beobachtung	1854 geometry lecture, opening plan: the empirical status of geometric assumptions.

Riemann is a name, not a claim. “Very probably” recalls the 1859 hypothesis; the game does not test or prove it. All raises share one dialogue pool, including bluff raises. Comments never consult hidden cards or the dealer’s equity score. Win, loss and split remarks appear only after the result is public.

Examples of original dialogue: “Let us test the foundations.” “A bold conjecture.” “A counterexample. Most instructive.” The stopping-short line is inspired by the end of the geometry lecture, but is not offered as a quotation. The complete dialogue and selection rule are published here.

Sources: Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (1859), German text above; English comparison: David R. Wilkins, 1998, p. 4. Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (1854 lecture), German text above; English comparison: William Kingdon Clifford, Nature 8 (1873), pp. 14–17, 36–37. The English bubble lines are our adaptations, not attributed to either translator.

Stamps are public information

If you have earned a card's stamp in Memory 51, that card shows its stamp on its back whenever it sits in Riemann's hand. The card's name is never shown — only the mark, so a remembered mapping can reveal information about the House hand. Riemann ignores marks entirely. The exact odds panel ignores them too, so it is a baseline before using stamp information. It reports your final hand category, not your chance of beating Riemann; no measured benefit of recall is claimed. An exact Standard-game initial-stamp information note derives the posterior and information gain under complete decoding; it does not assign a jbit value or change this panel.

What removing one card does to the odds

The exact counts over all C(52,5) = 2,598,960 and C(51,5) = 2,349,060 hands:

Five categories keep their probability exactly. One pair, two pair, three of a kind, full house and four of a kind are defined only by how often ranks repeat. Relabelling the ranks and, within each rank, the suits preserves every one of those categories, and those relabellings move any card to any other card. So every card lies in the same share of those hands — five in fifty-two — and removing one card keeps 47/52 of them, exactly the share of hands that survive overall. Their probability cannot change, and it does not.

Four categories move. A straight depends on the order of ranks, which relabelling does not preserve. The 2 appears in 2 of the 10 straight types, and the club is 1 of its 4 suits, so exactly 1/20 of straights and straight flushes are lost. The deck lost more than that — 9.6% of all hands — so both become about 5.11% more likely. Flush and high card probabilities fall by about 0.03999% relative to their 52-card values. These are relative changes, not percentage-point changes.

So a player using memorised 52-card odds is wrong on exactly four categories, and right on five to the last digit. The ranking of hands is unchanged.

Ⅰ back in play · Wild Ⅰ and the bug (research extension, v0.4)

The standard game keeps Ⅰ = 2♣ out of the deal. The game switch above the table offers two research variants that put it back, so 52 cards are in play. Every natural card keeps its meaning. A hand holding Ⅰ is worth the best hand it can make under the variant’s rule.

• Wild Ⅰ. Ⅰ stands for any card. With four of a rank it makes five of a kind, the new top hand.
• The bug. Ⅰ stands for an ace, or for any card that completes a straight, flush or straight flush. With four aces it makes five aces. This is the old draw-poker convention for a joker.

How Riemann adapts. Riemann always keeps Ⅰ and otherwise draws by the House rule applied to the hand Ⅰ makes at its best. Since v0.6 its betting in each variant comes from a table solved for that variant, with the variant’s own ranking and 52 cards: 28 dealt-hand classes for Wild Ⅰ and 36 for the bug (the standard classes, whether the hand holds Ⅰ, and its House-rule draw count), and 40 strength buckets. The figures are in the table under What the solve establishes.

Under v0.4 and v0.5 it played the standard policy with a belief table rebuilt for each variant, with seed 51 and the same sample sizes as the standard table. The rebuild mattered. With the standard table, about 1 in 22 of its after-draw decisions would have changed:

• Wild Ⅰ: 4.5% of 20,000 seeded spots. Its reads would have overstated its own hand by 2.4 points on average.
• The bug: 4.2% of 20,000 seeded spots, with almost no overall bias.

Before the draw, that policy projected its own draw from the 47 cards outside its hand.

Exact counts over every hand

Each variant counts C(52,5) = 2,598,960 hands, of which 249,900 hold Ⅰ. The standard game counts C(51,5) = 2,349,060.

1. Wild Ⅰ has no category order in which every higher category is rarer. Under the standard order, three of a kind (125,664 hands) is more common than two pair (111,672). Reversing those two categories sends 76,032 hands with one natural pair to two pair, making two pair more common instead. This is the known wild-card paradox (S. Gadbois, Mathematics Magazine 69, 1996) on this declared-card deck. A direct counting proof rules out every category order, even if equal frequencies are allowed; the search of all 3,628,800 orders is a separate check. The game still uses a fixed declared hierarchy, so each hand has a well-defined best reading and showdowns remain transitive.
2. The bug has one rarity-consistent category order, but can weaken a marker hand. Of all 3,628,800 orders, only the standard one (with five aces on top) ranks rarer categories higher. Compared with reading Ⅰ as ordinary 2♣, 69,277 of the 249,900 marker hands move up a category, 138,728 stay in the same category, and 41,895 move down. For example, Ⅰ, 2♦, 6♠, 9♥, Q♣ changes from a pair of twos to ace-high: the bug rule does not always allow Ⅰ to remain 2♣. These are category comparisons; hands in the same category can still differ in rank or kickers. The addendum gives the closed-form decline count. The playable preset is a design choice; returns and optimality have not been established.
3. Ⅰ as a two of any suit keeps five categories at their exact 52-card counts. Those are one pair, two pair, three of a kind, full house and four of a kind. Choosing Ⅰ’s suit never changes which ranks repeat, and a hand with a repeated rank can never be a flush. So each of those hands is in the same category as in the ordinary 52-card deck. Only 1,479 high-card hands become flushes, and 6 straights become straight flushes. This sits beside the removal theorem below: removing the card keeps the same five categories at their exact probabilities; making it suit-wild keeps them at their exact counts.
4. Ⅰ as a club of any rank makes full houses (6,678) more common than flushes (5,076). Of all 362,880 orderings, exactly one is consistent: the standard one with flush above full house, as short-deck poker ranks them.

Evidence: census (script) · every ranking checked (script) · verifier and its receipt · working note PP292 · rules poker51-variants-v1.js.

Working note · Riemann v0.6

Solving Riemann: the whole hand as one game · working note (unpublished; no DOI yet). It gives the abstraction, the solver, the player models, every score table and the limits.

Unpublished working paper

Poker 51 · v0.3 draft PDF, with pre-draw and post-draw betting · Editable text · earlier v0.1 draft. Prepared for the Project 51 collection; not deposited on Zenodo and no Poker-specific DOI is assigned. The paper includes exact counts, the invariance proof, the draw model and verification scope.

Status

Claim	Status	Where
Five-card counts on the 52- and 51-card decks	COMPUTED	every hand, two supplied evaluators and a separate C++ census; verifier · independent receipt
The five rank-multiplicity categories keep their probability exactly	PROVED	transitivity argument above; checked in integers by the verifier
Straights and straight flushes lose exactly 1/20	PROVED	2 of 10 straight types × 1 of 4 suits
The draw odds panel	COMPUTED	every draw from the 46 unseen cards, in your browser
Sampled two-round paths conserve jbits; selected draw and all-in rules hold	COMPUTED	v0.3 verifier · receipt; finite tests, not a formal proof
The belief table regenerates bit for bit from seed 51	COMPUTED	builder; verifier with --regenerate
The v0.6 table module plays the published tables exactly, replays byte for byte and keeps the v0.5 rules	COMPUTED	v0.6 verifier · receipt; 9,000 seeded hands, not a formal proof
Value of the whole hand, and what each v0.6 table guarantees, in the abstraction	COMPUTED	CFR+ on the stated abstraction; replay script · receipt
Riemann v0.6 wins more than v0.5 from four modelled player types on fresh real-card deals	EVID	the players are a language model’s answers; study
Strategy returns against real players, in any version	OPEN	not measured
Ⅰ any suit keeps pair, two pair, three of a kind, full house and four of a kind at their exact 52-card counts	PROVED	suit choice never changes repeated ranks; counts checked exactly in the census
Wild Ⅰ: no category order makes every higher category rarer, even allowing ties	PROVED; COMPUTED check	direct counting proof; all 3,628,800 orders in the search receipt
The bug and Ⅰ any suit admit only the standard rarity-consistent category order	COMPUTED	all 3,628,800 (or 362,880) orders; receipt
The bug moves 41,895 marker hands down a category relative to ordinary 2♣	PROVED	closed-form partition by number of natural twos
Riemann’s variant play, or either variant, is fair, optimal or an equilibrium	not claimed	its tables solve an abstraction of each variant; nothing is claimed for the card game
The House draw rule or Riemann's betting is optimal, or an equilibrium, in the real card game	not claimed	published fixed rules and tables; the solve is of an abstraction, with the House draw rule taken as given

Limits

Poker 51 runs entirely in your browser, including the House hand, so it is a strategy exhibit rather than a secure game: someone inspecting the page could read the deal. Hands are played for jbits in the shared Area 51 bank, which lives only in this browser and follows its normal recovery rules. Jbits have no monetary value and cannot be bought, sold or cashed out. A hand left open is saved with the bank and resumes where it stopped. Poker does not alter the stamp collection.

These are exact results about a card model. There is no zero data and no inference about the Riemann hypothesis.

Download the exact alphabet · Memory 51 · Blackjack 51 · Project 51: Reading the Record Across Rules · 10.5281/zenodo.23048482