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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)

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)

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.

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)

The game of nim, with its simple rules, its elegant solution and its historical importance is the quintessence of a combinatorial game, which is why it led to so many generalizations and modifications. We present a modification with a new spin: building nim. With given finite numbers of tokens and stacks, this two-player game is played in two stages (thus… (More)

The Tower of Hanoi game is a classical puzzle in recreational mathematics, which also has a strong record in pure mathematics. In a borderland between these two areas we find the characterization of the minimal number of moves, which is 2 n − 1, to transfer a tower of n disks. But there are also other variations to the game, involving for example move edges… (More)

- Urban Larsson, Nathan Fox
- 2015

In a recent manuscript, Fox studied infinite subtraction games with a finite (ternary) and aperiodic Sprague-Grundy function. Here we provide an elementary example of a game with the given properties, namely the game given by the subtraction set {F 2n+1 − 1}, where F i is the ith Fibonacci number, and n ranges over the positive integers.