QBF, which stands for Quantified Boolean Formula, is a two-player game. It’s played on an initially empty square board. Mark Steere, in collaboration with Claude, an AI model made by Anthropic, designed QBF in November 2026.
The two players, Red and Blue, take turns placing checkers of their own color onto unoccupied squares on the board. Red begins by placing a stack of two red checkers on any square. This stack is called the root, and is treated as a single, marked checker.
QBF uses the pie rule. Blue, for his first turn, can choose to switch colors with Red instead of placing a checker. If he does, the root remains red, but the player who placed the root is now Blue, and places a blue checker. Thereafter, players take turns placing one checker per turn.
After the root is placed, you must place your checker onto a square that is orthogonally adjacent to exactly one checker, of either color. Diagonal adjacencies are irrelevant.
In Figure 1, the root is outlined in bold. Dots mark the squares available to Blue. The checker that a new checker is placed next to is called its parent. The newly placed checker is a child of that parent. In Figure 2, each checker is joined to its parent by a line segment. The game ends when no unoccupied square is orthogonally adjacent to exactly one checker.
When the game ends, the players claim the checkers on the board. A player can claim ownership of checkers of either or both colors. Start with the checkers that have no children and work inward toward the root:
A checker belongs to its own color if all of its children belong to its own color. Otherwise it belongs to the other color. Childless checkers belong to their own color. You win if you claim the root.
In Figure 2, the game hasn’t ended yet, but the claiming process can be demonstrated nonetheless. The letter on each checker shows who currently owns it. Blue claims the red root, because Blue owns the checker above the root.
Multiple claims can change hands on every turn. In Figure 3, a continuation of Figure 2, Red has placed a checker above the blue checker two squares above the root. That blue checker now has a child owned by Red, so Red claims it. Red then claims the red checker above the root, and the root.
A paper I’m working on uses QBF in a proof. This gave me the idea of using QBF as the basis for a design. I had an initial idea, and Claude carried the idea farther from the tree.
Feel free to publish this rule sheet and to program the game of QBF. No licensing fee or royalties are expected. However, please don’t change the name or the rules, and please attribute the game to me, Mark Steere. My other games can be found at marksteeregames.com.
Red’s first move is the root: tap any square, and it is shown as a stack of two red checkers. After that the squares where you may place light up in green; tap one to place a checker. With the confirm step on, the checker is shown faintly on its square with a dashed gold outline until you tap Confirm, and Cancel takes it back. A white ring marks the last checker placed. With Show owners on, each checker carries a mark showing who owns it: a black ring when it belongs to its own color, and a disc of the other color when it belongs to the other color. With the confirm step on, the marks show the position after the move you are about to confirm. When the game ends, the marks are shown whatever the setting. When the root has been placed and Blue is a human, the app asks whether Blue wants to switch colors; a computer Blue switches when it judges the root square to favor Red. Undo takes back your move together with the computer’s reply, one move in two-player mode, and is greyed out when there is nothing to take back. Hint marks the move the strongest level would make. Record shows the moves of the current game as text you can copy. New game asks first when a game is in progress. Each side can be a human or a computer level, so you can play either color, play another person, or set both sides to the computer and watch the game play out.
Your settings and an unfinished game are kept in your browser.