This is the equilibrium of a simplified one-street river game, not a full hold’em solver. There is one fixed bet, followed by call or fold: no raises, future cards, ties, rake, or side pots. All value hands bet and beat this bluff-catcher. Air loses if it checks. The displayed combinations are conditional on the known cards and are equally weighted; all are interchangeable within their payoff class.

With pot P before the bet and bet B, a caller needs win probability B / (P + 2B). An indifferent air hand faces call frequency P / (P + B). If the starting range contains V value and A air combinations, the air range bets with aggregate frequency (V / A) × B / (P + B). The equilibrium drills use strictly interior cases, where that frequency lies between zero and one.

These frequencies describe a repeated randomized strategy, not a required action on a single deal. “Range EV” instead treats the displayed betting range as known and chooses the action with higher expected chips. Do not automatically defend at the equilibrium frequency against every opponent.

The formulas are derived from the payoffs of the basic endgame in Chris and Tom Ferguson’s The Endgame in Poker, section 1. Problems, explanations, and card graphics here are original. Scores reflect first submitted answers; revealing a solution earns no answer credit. No money is wagered.