Thirty-One is played with a standard 52-card deck against three computer players. Each player starts with 3 or 4 bitcoins and is dealt three cards. The goal each round is to avoid holding the lowest hand when the round ends.

Scoring

• Add the values of the cards you hold in one suit. Ace = 11, face cards = 10, other cards their number. Your score is your best single suit.
• 31 (Ace + two ten-value cards of one suit) wins the round instantly; everyone else loses a coin.
• Optional table rule: three cards of the same rank count 30½.

A turn

• Draw the top card from the deck, or take the face-up card from the discard pile, then put one card on the discard pile.
• Or, instead of drawing, knock. Every other player then gets one final turn, after which all hands are compared.

End of a round

• The lowest hand loses one bitcoin. If the knocker is lowest (or ties for lowest), the knocker loses 1 or 2 coins, as set at the table.
• A player with no coins is out. The last player with coins wins the game.

The Oracle

The Oracle is an advisor, not a boss. On your turn it leads with its best option and a plain-language reason (and says so when two options are statistically tied), then shows, for each of your options — knock, draw from the deck, take the discard — the percentage of simulated outcomes in which your hand was not the lowest after every remaining opponent took their final turn. Higher is safer. The decision stays yours.

How the Oracle calculates

• Monte Carlo simulation. Instead of a formula, it plays the round out thousands of times at random and counts the results. The simulation runs on the shared Monte Carlo engine used across tedponderings.com; this game supplies only its own rules.
• Seen and unseen cards. Your hand and every card that has passed through the discard pile are known; everything else is unseen. Only unseen cards are dealt in the simulation, so a buried card can never turn up in an opponent's simulated hand.
• Consistent opponents. The engine re-deals each opponent's hidden cards until they agree with everything that opponent visibly did this round — which discards they took, which cards they put down, and that they did not knock. A player who took the K♦ must have held diamonds worth taking it for. The test uses the game's own decision rules, including its tie-break for which card to put down.
• Playing it out. The engine collects up to 10,000 of those consistent deals and plays each option 50,000 times over them, with a fresh deck order each time, turn by turn until the showdown. Opponents play by the game's own rules, including knocking at 27 or more. If you draw or take while no one has knocked, the round goes on, so the Oracle does not guess when you would knock later: it plays the round once for every score at which you might knock (or never), keeps whichever does best, then plays that choice another 50,000 times to confirm it.
• Precision. The estimate carries a 95% confidence margin of 1.96 × √(p(1−p)/n), where n is the number of consistent deals, about ±1 point. Two options within that margin of each other are statistically equal, and the Oracle says so.