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Why the thresholds are exactly three
Not arbitrary — it falls out of the arithmetic of a five-card hand built from two hole cards.
Straights need three board cards because a straight is five ranks in a row and you only hold two cards. Three from the board plus your two is the minimum. So two connected cards on the board — even 9-8 sitting right there — cannot make anyone a straight. It's a warning about the next card, not a present danger.
Flushes need three of a suit for the same reason: three on the board plus two in your hand. And the count changes what your opponent needs:
| Cards of one suit on board | What a player needs | Practical meaning |
|---|---|---|
| 2 | nothing yet | no flush exists — it's a draw, not a hand |
| 3 | both hole cards of that suit | possible but uncommon |
| 4 | one hole card of that suit | likely — someone usually has it |
| 5 | nothing — everyone plays the board | the highest card of that suit in anyone's hand wins |
Full houses need a paired board — otherwise a full house would require three of your two cards. When the board is unpaired, full houses and quads are simply off the menu, no matter how many people are in the pot.