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

The switch operators and push-the-button games: a sequential compound over rulesets

We study operators that combine combinatorial games. This field was initiated by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s) via the now classical disjunctive sum operator on (abstract) games. The new class consists in operators for rulesets, dubbed the switch-operators. The ordered pair of rulesets (R 1 , R 2) is compatible if, given any position in R 1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2), we build the push-the-button game R 1 R 2 , where players start by playing according to the rules R 1 , but at some point during play, one of the players must switch the rules to R 2 , by pushing the button ". Thus, the game ends according to the terminal condition of ruleset R 2. We study the pairwise combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we prove that standard periodicity results for Subtraction games transfer to this setting, and we give partial results for a variation of Domineering, where R 1 is the game where the players put the domino tiles horizontally and R 2 the game where they play vertically (thus generalizing the octal game 0.07).Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A Para{\^i}tr

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