Picture-Hanging Puzzles

We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.Comment: 18 pages, 8 figures, 11 puzzles. Journal version of FUN 2012 pape

Picture-hanging puzzles

We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.National Science Foundation (U.S.) (NSF grant CCF-1018388

Simple realizability of complete abstract topological graphs simplified

An abstract topological graph (briefly an AT-graph) is a pair $A=(G,\mathcal{X})$ where $G=(V,E)$ is a graph and $\mathcal{X}\subseteq {E \choose 2}$ is a set of pairs of its edges. The AT-graph $A$ is simply realizable if $G$ can be drawn in the plane so that each pair of edges from $\mathcal{X}$ crosses exactly once and no other pair crosses. We show that simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent $\mathbb{Z}_2$-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.Comment: 26 pages, 17 figures; major revision; original Section 5 removed and will be included in another pape

Bulletin of the Center for Children's Books 20 (06) 1967

published or submitted for publicatio

Player identification in American McGee’s Alice : a comparative perspective

In this paper I analyse personal identification in three incarnations of Alice in Wonderland: the original novels, the 1950s Disney animation film and the computer game American McGee’s Alice. After presenting the research corpus, I lay out the analytical framework derived from Kendall Walton’s theory of representational artefacts as props for evoking imagining in games of make-believe. From this perspective, the Alice heritage relies on spectacle rather than plot to entertain. This spectacle differs across media as each medium’s strengths are played out: language-play in the novels, colour/motion/sound in the film and challenges in the game. There are two types of imagining involved: objective, whereby a person imagines a scene outside of himself, and subjective, in which case the imagining revolves around a version of himself. Both the novels and the film primarily evoke objective imagining whereas the game invites the player to be introjected into the Alice character evoking subjective imagining. The picture is not unambiguous, however, as the novels and the film stage a broad array of subjectifying techniques and the game objectifying ones. This gives us some indication as to the nature of representation which, to be of interest, presents a tension between here and there, between the self and an other

Large Peg-Army Maneuvers

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Cs\'ak\'any and Juh\'asz (Mathematics Magazine, 2000). To appreciate the combinatorial beauty of our solutions, we recommend to visit the gallery of animations provided at http://solitairearmy.isnphard.com.Comment: Conference versio

High spin baryon in hot strongly coupled plasma

We consider a strings-junction holographic model of probe baryon in the finite-temperature supersymmetric Yang-Mills dual of the AdS-Schwarzschild black hole background. In particular, we investigate the screening length for high spin baryon composed of rotating N_c heavy quarks. To rotate quarks by finite force, we put hard infrared cutoff in the bulk and give quarks finite mass. We find that N_c microscopic strings are embedded reasonably in the bulk geometry when they have finite angular velocity \omega, similar to the meson case. By defining the screening length as the critical separation of quarks, we compute the \omega dependence of the baryon screening length numerically and obtain a reasonable result which shows that baryons with high spin dissociate more easily. Finally, we discuss the relation between J and E^2 for baryons.Comment: 18 pages, 19 figures, version to appear in JHE

The puzzles of topic dislocations in Polish

Niniejszy artykuł analizuje konstrukcje, w których uwydatniona jest informacja dana. W pierwszej części artykułu dokonujemy klasyfikacji topików znajdujących się w analizowanych konstrukcjach. Ponieważ proces topikalizacji kojarzony jest zazwyczaj z przesunięciem w strukturze, wybrane testy pokazują czy rzeczywiście wszystkie badane konstrukcje wykazują cechy przesunięcia. Jak się okazuje, konstrukcje z topikiem kontrastywnym potrzebują analizy, która pogodzi ze sobą cechy przesunięcia do pozycji argumentu oraz do pozycji innej niż pozycja argumentu. Przedstawione dowody świadczą o tym, iż ruch dopełnienia w konstrukcjach o szyku dopełnienie-to-podmiot-orzeczenie jest ruchem niekwantyfikującym i wieloetapowym: najpierw do pozycji nad podmiotem aczkolwiek wykazującej cechy podmiotu, a następnie do pozycji okupowanej przez topiki.In this paper we show that there are different topic dislocations in Polish, each representing a specific type of a discourse function. With a battery of diagnostic tests we analyse each dislocation and propose their classification. As it turns out, constructions implementing a contrastive topic exhibit features of both A and A'-movement, which turns out problematic for a uniform analysis. We demonstrate that the movement in them is non-quantificational. The movement targeting TopP consists of at least two steps. An object undergoes A-movement and lands in the specifier of an Aboutness Phrase. Then it moves to SpecTopP where it checks a discourse feature