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The game is played between two players on a board consisting of whole numbered tokens labeled 1 through N, where N is any positive whole number. During each turn, one player (deemed the tax payer) takes a number from the board, and the other player (deemed the taxman) removes all remaining factors of the tax payer's number from the board.
The higher the branching factor, the faster this "explosion" occurs. The branching factor can be cut down by a pruning algorithm. The average branching factor can be quickly calculated as the number of non-root nodes (the size of the tree, minus one; or the number of edges) divided by the number of non-leaf nodes (the number of nodes with ...
The transformations of the 15 puzzle form a groupoid (not a group, as not all moves can be composed); [12] [13] [14] this groupoid acts on configurations.. Because the combinations of the 15 puzzle can be generated by 3-cycles, it can be proved that the 15 puzzle can be represented by the alternating group. [15]
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The game tree size is the total number of possible games that can be played: the number of leaf nodes in the game tree rooted at the game's initial position.. The game tree is typically vastly larger than the state space because the same positions can occur in many games by making moves in a different order (for example, in a tic-tac-toe game with two X and one O on the board, this position ...
The total number of possible games can be estimated from the board size in a number of ways, some more rigorous than others. The simplest, a permutation of the board size, (N) L, fails to include illegal captures and positions. Taking N as the board size (19 × 19 = 361) and L as the longest game, N L forms an upper limit. A more accurate limit ...
The complete game tree for a game is the game tree starting at the initial position and containing all possible moves from each position; the complete tree is the same tree as that obtained from the extensive-form game representation. To be more specific, the complete game is a norm for the game in game theory.
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