Heap games, numeration systems and sequences

We propose and analyse a 2-parameter family of 2-player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an efficient strategy or not. We introduce yet another class of sequences, and demonstrate its equivalence with the class of sequences defined for the strategy of our games.Comment: To appear in Annals of Combinatoric

When are translations of P-positions of Wythoff's game P-positions?

We study the problem whether there exist variants of {\sc Wythoff}'s game whose $\P$-positions, except for a finite number, are obtained from those of {\sc Wythoff}'s game by adding a constant $k$ to each $\P$-position. We solve this question by introducing a class \{\W_k\}_{k \geq 0} of variants of {\sc Wythoff}'s game in which, for any fixed $k \geq 0$, the $\P$-positions of \W_k form the set $\{(i,i) | 0 \leq i < k\}\cup \{(\lfloor \phi n \rfloor + k, \lfloor \phi^2 n \rfloor + k) | n\ge 0\}$, where $\phi$ is the golden ratio. We then analyze a class \{\T_k\}_{k \geq 0} of variants of {\sc Wythoff}'s game whose members share the same $\P$-positions set $\{(0,0)\}\cup \{(\lfloor \phi n \rfloor + 1, \lfloor \phi^2 n \rfloor + 1) | n \geq 0 \}$. We establish several results for the Sprague-Grundy function of these two families. On the way we exhibit a family of games with different rule sets that share the same set of $\P$-positions

Planar kernel and grundy with d≤3, dout≤2, din≤2 are NP-complete

AbstractIt is proved that the questions whether a finite diagraph G has a kernel K or a Sprague—Grundy function g are NP-complete even if G is a cyclic planar digraph with degree constraints dout(u)≤2, din(u)≤2 and d(u)≤3. These results are best possible (if P ≠ NP) in the sense that if any of the constraints is tightened, there are polynomial algorithms which either compute K and g or show that they do not exist. The proof uses a single reduction from planar 3-satisfiability for both problems

How far can Nim in disguise be stretched?

A move in the game of nim consists of taking any positive number of tokens from a single pile. Suppose we add the class of moves of taking a nonnegative number of tokens jointly from all the piles. We give a complete answer to the question which moves in the class can be adjoined without changing the winning strategy of nim. The results apply to other combinatorial games with unbounded Sprague-Grundy function values. We formulate two weakened conditions of the notion of nim-sum 0 for proving the results.Comment: To appear in J. Combinatorial Theory (A

Scenic trails ascending from sea-level Nim to alpine chess

Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Ascend from Nim to chess in small strides at a gradient that's not too steep. Presentation: Informal; examples of games sampled from various strategic viewing points along scenic mountain trails, which illustrate the theory.Comment: 25 page

Invariant and dual subtraction games resolving the Duch\^e-Rigo conjecture

We prove a recent conjecture of Duch\^ene and Rigo, stating that every complementary pair of homogeneous Beatty sequences represents the solution to an \emph{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 sequences, and in the process generalize the notion of a \emph{subtraction game}. Given a pair of complementary sequences $(a_n)$ and $(b_n)$ of positive integers, we define a game $G$ by setting $\{\{a_n, b_n\}\}$ as invariant moves. We then introduce the invariant game $G^\star$, whose moves are all non-zero $P$-positions of $G$. Provided the set of non-zero $P$-positions of $G^\star$ equals $\{\{a_n,b_n\}\}$, this \emph{is} the desired invariant game. We give sufficient conditions on the initial pair of sequences for this 'duality' to hold.Comment: 11 pages, 2 figure