# Sequences and games generalizing the combinatorial game of Wythoff Nim Licentiate thesis, 2009

One single Queen is placed on an arbitrary starting position of a (large) Chess board. Two players alternate in moving the Queen as in a game of Chess but with the restriction that the $L^1$ distance to the lower left corner, position $(0,0)$, must decrease. The player who moves there wins. Let $\phi =\frac{1+\sqrt{5}}{2}$, the golden ratio. In 1907 W. A. Wythoff proved that the second player wins if and only if the coordinates of the starting position are of the form $\{a_n,b_n\}$, where $a_n=\left\lfloor n\phi \right\rfloor, b_n=a_n+n$ for some non-negative integer $n$. Here, we introduce the game of \emph{Imitation Nim}, a \emph{move-size dynamic} restriction on the classical game of (2-pile) Nim. We prove that this game is a 'dual' of \emph{Wythoff Nim} in the sense that the latter has the same solution/$P$-positions as the former. On the one hand we define extensions and restrictions to Wythoff Nim---including the classical generalizations by I.G. Connell (1959) and A.S. Fraenkel (1982)---and Imitation Nim. All our games are purely combinatorial, so there are no 'hidden cards' and no 'chance device'. In fact we only study so-called \emph{Impartial games} where the set of options does not depend on whose turn it is. In particular we introduce rook-type and bishop-type \emph{blocking manoeuvres/Muller twists} to Wythoff Nim: For each move, the previous player may 'block off' a predetermined number of next player options. We study the solutions of the new games and for each blocking manoeuvre give non-blocking dual game rules. On the other hand, observing that the pair of sequences $(a_n)$ and $(b_n)$---viewed as a permutation of the natural numbers which takes $a_n$ to $b_n$ and $b_n$ to $a_n$---may be generated by a 'greedy' algorithm, we study extensive generalizations to these. We also give interpretations of our sequences as so-called \emph{Interspersion arrays} and/or \emph{Beatty sequences}.

Beatty sequence

Interspersion array

Blocking manoeuvre

Impartial game

Complementary sequences

Wythoff Nim

Combinatorial game

Nim

Stolarsky array

Muller twist

Permutation of the natural numbers

Euler
Opponent: Prof. Aviezri S. Fraenkel, Department of Computer Science & Applied Mathematics, Weizmann Institute of Science, Israel

## Author

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

### Permutations of the Natural Numbers with Prescribed Difference Multisets

INTEGERS : The Electronic Journal of Combinatorial Number Theory,; Vol. 6(2006)p. paper A 3-

Journal article

### Subject Categories

Discrete Mathematics

Preprint - Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University: 2009:42

Euler

Opponent: Prof. Aviezri S. Fraenkel, Department of Computer Science & Applied Mathematics, Weizmann Institute of Science, Israel