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

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 Hut on the Garden Plot - Informal Architecture in Twentieth-Century Berlin

In Berlin, self-built huts and sheds were a part of the urban fabric for much of the twentieth century. They started to proliferate after the First World War and were particularly common after the Second World War, when many Berliners had lost their homes in the bombings. These unplanned buildings were, ironically, connected to one of the icons of German orderliness: the allotment. Often depicted as gnome-adorned strongholds of petty bourgeois virtues, garden plots were also the site of mostly unauthorized architecture and gave rise to debates about public health and civic order. This paper argues that the evolution and subsequent eradication of informal architecture was an inherent factor in the formation of the modern, functionally separated city. Modern Berlin evolved from a struggle between formal and informal, regulation and unruliness, modernization and pre-modern lifestyles. In this context, the ambivalent figure of the allotment dweller, who was simultaneously construed as a dutiful holder of rooted-to-the-soil values and as a potential threat to the well-ordered urban environment, evidences the ambiguity of many conceptual foundations on which the modern city was built