Sylver

Sylver is a combinatorial game played between two players, traditionally called A and B. The players alternate turns selecting positive integers. A player cannot select an integer that can be expressed as a sum of previously chosen integers (with repetition allowed). The player who cannot make a valid move loses.

More formally, the first player, A, chooses a positive integer. Then the second player, B, chooses a positive integer different from A's choice. Play continues, with each player choosing a positive integer that is not in the set of linear combinations (with non-negative integer coefficients) of the integers already chosen. The game ends when every positive integer is either in the set of chosen integers or can be expressed as a sum of previously chosen integers. The winner is the last player to make a valid move.

The integers chosen form a set, often denoted as S. The set of all positive integers expressible as a non-negative integer linear combination of the elements of S is denoted <S>. Thus, a player can only choose an integer not in <S>.

The game is known to always terminate; one player must eventually run out of valid moves. Determining optimal strategies in Sylver is a complex problem, and the game is still the subject of mathematical research. Understanding the properties of the sets S that lead to a win or a loss for the first player is a key area of study.