OSLR

Open Solving the Last-Round Problem, often abbreviated as OSLR, refers to a specific challenge within the domain of combinatorial game theory, particularly in relation to impartial games. An impartial game is one where the available moves from any position depend only on the position itself, not on whose turn it is. Examples include Nim and Dawson's Kayles.

The "Last-Round Problem" typically arises when considering games that can be decomposed into a sum of simpler games, a common technique in combinatorial game theory. The sum of games means that a player can choose to move in any one of the component games. The game ends when no moves are possible in any of the component games. The player who makes the last move wins (normal play convention).

OSLR specifically addresses the situation where one of the component games in a sum has a very limited number of possible moves in its final stages. This scarcity of moves can complicate the standard analysis methods used for sums of games, such as the Sprague-Grundy theorem, which relies on calculating the nim-value (or Grundy number) of each component game. The nim-value of a game position is the smallest non-negative integer that is not among the nim-values of the positions reachable in one move. The nim-value of a sum of games is the bitwise XOR sum of the nim-values of its components.

The "Open Solving" aspect implies that there isn't a universally accepted or straightforward algorithm to solve this specific type of problem for all impartial games. Researchers and players of combinatorial games often need to devise tailored strategies or computational approaches to handle situations where the last few moves of a component game are difficult to analyze within the standard framework. This can involve detailed case analysis or specialized algorithms for the game in question.