Poker 51.
Research ↗One card is declared. The House plays a published rule. You play the odds — and whatever you remember.
Your draw, exactly · the 51-card odds
Record · receipts
Research · Rule Card / Live Paper
POKER 51 / LIVE RULE CARD · v0.1
One card declared. One rule published.
Poker 51 is heads-up five-card draw on the 51-card Area 51 deck. The marker Ⅰ = 2♣ is outside the deal, and both players know it. That is the only card excluded from play. Earned stamps may also identify face-down House cards to a player who remembers their mapping. The deck is shuffled fresh for every hand with your browser's cryptographic random source.
A hand
- You and the House each receive five cards. Yours are face up to you; the House's are face down.
- You choose which cards to keep and draw the rest from the deck — any number from none to five.
- The House then keeps and draws by its published rule. How many cards it drew is shown, as it would be at a real table.
- You show or fold. On a show, the better five-card hand wins under standard poker ranking; equal hands split. If you fold, the House hand is still revealed, so you can compare the hands. With no stakes, “good fold” only means your hand would have lost at showdown.
The House rule
The House plays like a dealer: its own five cards and this rule, nothing else. It never reads stamps and never sees your hand.
- Straight or better: keep all five.
- Three of a kind: keep the three.
- Two pair: keep both pairs.
- One pair: keep the pair.
- Four to a flush: keep the four.
- Four to an open-ended straight — four consecutive ranks from 2-3-4-5 up to 10-J-Q-K: keep the four.
- Otherwise keep the single highest card; a tie goes to the suit order ♠ ♥ ♦ ♣.
This is a simple, fixed rule. Its optimality and a player’s achievable win rate have not been established. Because the rule is public, proposed strategies can be tested against it.
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 the House hand. The card's name is never shown — only the mark, so a remembered mapping can reveal information about the House hand. The House 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 the House; no measured benefit of recall is claimed.
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.
Unpublished working paper
Poker 51: One Card Declared, Five Probabilities Preserved · v0.1 draft PDF · Editable text. 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 |
| The House rule is optimal | not claimed | it is a published fixed rule |
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. Poker actions place no wagers, pay no awards and do not alter the stamp collection in v0.1. Reading stamps uses the existing shared wallet service, which can initialize or refresh the bank under its normal recovery rules. Your record is kept only in this browser. There is no stake and nothing of monetary value.
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.23004789