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

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

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

**Journal article**

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