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Penney's game, named after its inventor Walter Penney, is a binary (head/tail) sequence generating game between two players. Player A selects a sequence of heads and tails (of length 3 or larger), and shows this sequence to player B. Player B then selects another sequence of heads and tails of the same length. Subsequently, a fair coin is ...
Consider the sum of the two games. The claim that G 1 and G 2 have the same Sprague-Grundy value is equivalent to the claim that the sum of the two games has Sprague-Grundy value 0. In other words, we are to show that the sum G 1 + G 2 is a P-position. A player is guaranteed to win if they are the second player to move in G 1 + G 2.
The goal of each player is to either advance a pawn to the opposite end of the board or leave the other player with no legal moves, either by stalemate or by having all of their pieces captured. Hexapawn on the 3×3 board is a solved game ; with perfect play, White will always lose in 3 moves (1.b2 axb2 2.cxb2 c2 3.a2 c1#).
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An arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications, such as the computation of expected values in probability theory , especially in Bernoulli processes .
Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2. Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is , , , , , … where r is the common ratio and a is the initial value. The sum of a geometric progression's terms is ...
The first player (or team) to complete two sequences of five chips in a row successfully, whether vertically, horizontally, or diagonally, wins the game of Sequence. A note about special cards
Wythoff's game is played with two piles of counters. Wythoff's game is a two-player mathematical subtraction game, played with two piles of counters.Players take turns removing counters from one or both piles; when removing counters from both piles, the numbers of counters removed from each pile must be equal.