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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)

If k is a positive integer, we say that a set A of positive integers is k-sum-free if there do not exist a, b, c in A such that a + b = kc. In particular we give a precise characterization of the structure of maximum sized k-sum-free sets in {1,. .. , n} for k ≥ 4 and n large.

- Urban Larsson
- 2008

We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint: Provided the previous player has just removed say x > 0 tokens from the pile with less tokens, the next player may remove x tokens from the pile with more tokens. But for each move, in " a strict sequence of previous player-next player… (More)

- Urban Larsson
- 2009

Fix a positive integer m. The game of m-Wythoff Nim (A.S. Fraenkel, 1982) is a well-known extension of Wythoff Nim (W.A. Wythoff, 1907). The set of P-positions may be represented as a pair of increasing sequences of non-negative integers. It is well-known that these sequences are so-called complementary Beatty sequences, that is they satisfy Beatty's… (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. (and where xi − mi ≥ 0, for all i). Two players alternate in moving and the player who moves last wins. The set of non-zero P-positions of the game M defines the moves in the dual game M ⋆. For example, in the game of (2-pile… (More)

We study permutations π of the natural numbers for which the numbers π(n) are chosen greedily under the restriction that the differences π(n)−n belong to a given (multi)subset M of Z for all n ∈ S, a given subset of N. Various combinatorial properties of such permutations (for quite general M and S) are exhibited and others conjectured. Our results… (More)

- Urban Larsson
- 2011

The P-positions of the well-known 2-pile takeaway 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)

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)

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)

- Urban Larsson, Mike Weimerskirch
- 2011

We study 2-player impartial games of the form takeaway which produce P-positions (second player winning positions) corresponding to so-called complementary Beatty sequences, given by the continued Our problem is the opposite of the main field of research in this area, which is to, given a game, understand its set of P-positions. We are rather given a set of… (More)