Urban Larsson

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We prove a recent conjecture of Duchêne and Rigo, stating that every complementary pair of homogeneous Beatty sequences represents the solution to an invariant impartial game. Here invariance means that each available move in a game can be played anywhere inside the game board. In fact, we establish such a result for a wider class of pairs of complementary(More)
The P-positions of the well-known 2-pile take-away game of Wythoff Nim lie on two ‘beams’ of slope √ 5+1 2 and √ 5−1 2 respectively. We study extensions to this game where a player may also remove simultaneously pt tokens from either of the piles and qt from the other, where p < q are given positive integers and where t ranges over the positive integers. We(More)
The 2-player impartial game of Wythoff Nim is played on two piles of tokens. A move consists in removing any number of tokens from precisely one of the piles or the same number of tokens from both piles. The winner is the player who removes the last token. We study this game with a blocking maneuver, that is, for each move, before the next player moves the(More)
An invariant subtraction game is a 2-player impartial game defined by a set of invariant moves (k-tuples of non-negative integers) M. Given a position (another k-tuple) x = (x1, . . . , xk), each option is of the form (x1 − m1, . . . , xk − mk), where m = (m1, . . . ,mk) ∈ M (and where xi − mi ≥ 0, for all i). Two players alternate in moving and the player(More)
We study so-called invariant games played with a fixed number d of heaps of matches. A game is described by a finite list M of integer vectors of length d specifying the legal moves. A move consists in changing the current game-state by adding one of the vectors in M, provided all elements of the resulting vector are nonnegative. For instance, in a two-heap(More)
We study two-player take-away games whose outcomes emulate two-state one-dimensional cellular automata, such as Wolfram’s rules 60 and 110. Given an initial string consisting of a central data pattern and periodic left and right patterns, the rule 110 cellular automaton was recently proved Turing-complete by Matthew Cook. Hence, many questions regarding its(More)