Blocking Wythoff Nim

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 previous player may declare at most a predetermined number, $k - 1 \ge 0$, of the options as forbidden. When the next player has moved, any blocking maneuver is forgotten and does not have any further impact on the game. We resolve the winning strategy of this game for $k = 2$ and $k = 3$ and, supported by computer simulations, state conjectures of the asymptotic `behavior' of the $P$-positions for the respective games when $4 \le k \le 20$.Comment: 14 pages, 1 Figur

2-pile Nim with a Restricted Number of Move-size Imitations

We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint: Suppose the previous player has just removed say $x>0$ tokens from the shorter pile (either pile in case they have the same height). If the next player now removes $x$ tokens from the larger pile, then he imitates his opponent. For a predetermined natural number $p$, by the rules of the game, neither player is allowed to imitate his opponent on more than $p-1$ consecutive moves. We prove that the strategy of this game resembles closely that of a variant of Wythoff Nim--a variant with a blocking manoeuvre on $p-1$ diagonal positions. In fact, we show a slightly more general result in which we have relaxed the notion of what an imitation is.Comment: 18 pages, with an appendix by Peter Hegart

Impartial games emulating one-dimensional cellular automata and undecidability

We study two-player \emph{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 behavior are algorithmically undecidable. We show that similar questions are undecidable for our \emph{rule 110} game.Comment: 22 pages, 11 figure

Restrictions of mm-Wythoff Nim and pp-complementary Beatty Sequences

Fix a positive integer $m$. The game of \emph{$m$-Wythoff Nim} (A.S. Fraenkel, 1982) is a well-known extension of \emph{Wythoff Nim}, a.k.a 'Corner the Queen'. Its set of $P$-positions may be represented by a pair of increasing sequences of non-negative integers. It is well-known that these sequences are so-called \emph{complementary homogeneous} \emph{Beatty sequences}, that is they satisfy Beatty's theorem. For a positive integer $p$, we generalize the solution of $m$-Wythoff Nim to a pair of \emph{$p$-complementary}---each positive integer occurs exactly $p$ times---homogeneous Beatty sequences a = (a_n)_{n\in \M} and b = (b_n)_{n\in \M}, which, for all $n$, satisfies $b_n - a_n = mn$. By the latter property, we show that $a$ and $b$ are unique among \emph{all} pairs of non-decreasing $p$-complementary sequences. We prove that such pairs can be partitioned into $p$ pairs of complementary Beatty sequences. Our main results are that \{\{a_n,b_n\}\mid n\in \M\} represents the solution to three new '$p$-restrictions' of $m$-Wythoff Nim---of which one has a \emph{blocking maneuver} on the \emph{rook-type} options. C. Kimberling has shown that the solution of Wythoff Nim satisfies the \emph{complementary equation} $x_{x_n}=y_n - 1$. We generalize this formula to a certain '$p$-complementary equation' satisfied by our pair $a$ and $b$. We also show that one may obtain our new pair of sequences by three so-called \emph{Minimal EXclusive} algorithms. We conclude with an Appendix by Aviezri Fraenkel.Comment: 22 pages, 2 figures, Games of No Chance 4, Appendix by Aviezri Fraenke

The ⋆\star-operator and Invariant Subtraction Games

We study 2-player impartial games, so called \emph{invariant subtraction games}, of the type, given a set of allowed moves the players take turn in moving one single piece on a large Chess board towards the position $\boldsymbol 0$. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, $\star$ of the old game, by taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One such game is \W^\star= (Wythoff Nim)$^\star$, where the moves are defined by complementary Beatty sequences with irrational moduli. Here we give a polynomial time algorithm for infinitely many $P$-positions of \W^\star. A repeated application of $\star$ turns out to give especially nice properties for a certain subfamily of the invariant subtraction games, the \emph{permutation games}, which we introduce here. We also introduce the family of \emph{ornament games}, whose $P$-positions define complementary Beatty sequences with rational moduli---hence related to A. S. Fraenkel's `variant' Rat- and Mouse games---and give closed forms for the moves of such games. We also prove that ($k$-pile Nim)$^{\star\star}$ = $k$-pile Nim.Comment: 30 pages, 5 figure