When to stop flipping in Flip 7

Hypotheses

1. Count. There is an ideal number of cards to stay at.
2. Sum. The right stop is a point total, not a card count.
3. Counting. Watching what others have flipped this round improves the decision.
4. Players. The best rule depends on how many people are at the table.
5. The table. The best rule depends on how the others play, and a table of good players settles somewhere.

Method

A simulation of the full game: the 94-card deck (one 0, then as many of each number as its value, up to twelve 12s), +2 to +10 and ×2, three each of Freeze, Flip Three and Second Chance; the deal; busts; Flip 7 ending the round for everyone with +15; discards reshuffled when the pile runs out; games to 200. Strategies only decide hit or stay. Who gets a Freeze or a Flip Three follows one house rule for everyone (freeze the opponent with the most on the table; dump a Flip Three on anyone holding four or more cards, otherwise take it), so rules are compared on the decision alone.

Four families of rule, each in several strengths:

• cards ≥ n: stay once you hold n numbers.
• sum ≥ s: stay once your banked value reaches s.
• bust ≤ p: hit while the chance the next card busts you is at most p.
• EV: hit when one more card raises the expected round score; the seventh card carries the +15. Variants count only your own hand, the whole table this round, or the exact pile. A race add-on locks in a win when staying wins the game and keeps hitting when staying would lose it.

Alone: 20,000 rounds per rule. In games: 2,000 per cell, one seat on the rule under test, the rest on a fixed field rule, the seat rotating so nobody always deals. Ties count half. The code is in the site repo under src/lib/flip7.

The decision at the table

The chance the next card busts you is the number of twins left in the unseen pile divided by the unseen pile. That depends on which cards you hold, not how many: a lone 12 has eleven twins out, a lone 2 has one. Hitting pays when the expected gain times the odds of surviving beats the bust times what you would bank. Build a hand and see.

Results

H1 · Count

cards≥3 · 18.3 avg · busts 24%

cards≥5 · 15.4 avg · busts 55%

cards≥7 · 9.7 avg · busts 74%

EV · 20.1 avg · busts 29%

Three and four cards tie alone (18.3 and 18.1 points a round) and everything beyond four is worse. Chasing Flip 7 gets it 9% of rounds and averages 9.7 points, because it busts 74% of the time. A count is the weakest kind of rule: it ignores what the cards are.

H2 · Sum

A sum target does a little better than any count (sum ≥ 25: 18.9), a bust cap does better again (bust ≤ 20%: 19.9), and the EV rule is best (20.1). The ordering is the information each rule uses: a count knows nothing about the hand, a sum knows the value, a bust cap knows the risk, and EV weighs both.

H3 · Counting

Reading the table is worth about half a point of win rate at a four-player table; knowing the whole pile, about a point and a half. The deck is 94 cards and a round exposes a few dozen, so the odds barely move. Game awareness is worth more than either.

H4 · Players

Everyone else playscards≥4cards≥5sum≥20sum≥30EV

• cards≥3
• cards≥4
• cards≥5
• sum≥20
• sum≥30
• bust≤20%
• EV
• EV + race

The best rule is the same at every table size. What changes is the price of caution: stay-at-3 takes 39% of two-player games (fair is 50%) and 8% of six-player games (fair is 17%), while stay-at-4, worse alone, beats it from three players up. More opponents means more chances someone posts a big round, so a low, safe score stops winning.

H5 · The table

Against a cautious table (everyone stays at 20) the EV rule wins half the games at four players. Against a table of EV-with-race, the best answer in the catalogue is EV-with-race itself, at a fair 24%. Within these families it is an equilibrium: once the table plays it, nothing here exploits it.

Try it

Any rule against any rule, as many players as you like; the win rate settles as the games run.

Or sit down against three bots that play the equilibrium rule, with the odds shown each turn.

Conclusions

1. There is no magic number of cards. Three and four tie alone; in games four edges three; past four it costs points.
2. The stop should come from the hand, not the count. Hit while the expected gain beats the expected loss on the next card, with the +15 priced in on the seventh.
3. Counting barely helps. Half a point for the table, a point and a half for the whole pile.
4. Table size does not change the rule, only the penalty for playing safe, which grows with every extra player.
5. Watch the scoreboard. Endgame awareness is worth two points of win rate, more than counting, and the EV-plus-race rule is where a table of good players settles.

Caveats

The EV rule looks one card ahead; a full dynamic program over every possible pile would do slightly better. Freeze and Flip Three targets are a house rule rather than a strategy, and people play those cards with spite. The bots never bluff or read faces, and real players do not hold a fixed rule for a whole game. Next step: log real games in the score tracker and see how far the table at home sits from the equilibrium.