Related papers: How do you fix an Oval Track Puzzle?

For each $n\leq 6$, we characterize all the groups which can occur as either the orientation preserving topological symmetry group or the topological symmetry group of some embedding of $K_n$ in $S^3$.

We prove the computational intractability of rotating and placing $n$ square tiles into a $1 \times n$ array such that adjacent tiles are compatible--either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as…

Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling…

A basic problem in spectral clustering is the following. If a solution obtained from the spectral relaxation is close to an integral solution, is it possible to find this integral solution even though they might be in completely different…

Let T(N) be the subgroup of the mapping class group of a nonorientable surface N (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for T(N). As an application…

The tiling problem has been a famous problem that has appeared in many Mathematics problems. Many of its solutions are rooted in high-level Mathematics. Thus we hope to tackle this problem using more elementary Mathematics concepts. In this…

Tetravex is a widely played one person computer game in which you are given $n^2$ unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a $n$ by $n$ square such that all neighbouring edges are…

This paper addresses the problem of finding $Q_{m,t}\left(n\right)$, the number of possible ways to partition any member $n$ of the cyclic group $\mathbb{Z}/m\mathbb{Z}$ into $t$ distinct parts. When $m$ is odd, it was previously known that…

Consider a compact oriented surface $S$ of genus $g \geq 0$ and $m \geq 0$ punctured. The train track complex of $S$ which is defined by Hamenst\"adt is a 1-complex whose vertices are isotopy classes of complete train tracks on $S$.…

The orthogonal beltway problem is the problem of recovering the $\mathrm{O}(n)$-orbit of a $\delta$-function supported at a finite number of points in $\r^n$ from its auto-correlation or, equivalently, second moment. It was introduced as a…

In the trace reconstruction problem, one seeks to reconstruct a binary string $s$ from a collection of traces, each of which is obtained by passing $s$ through a deletion channel. It is known that $\exp(\tilde O(n^{1/5}))$ traces suffice to…

An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile'". The tile may or may not be similar to ABC . This paper is the…

Origami is the art of folding paper into various patterns without cutting or tearing the paper. By viewing the paper as the complex plane, we iteratively compute and record all intersection points to construct mathematical origami sets.…

Satisfiability is a classic problem in computational complexity theory, in which one wishes to determine whether an assignment of values to a collection of Boolean variables exists in which all of a collection of clauses composed of logical…

For nonempty subsets $X$ and $Y$ of a group $G$, we say that $(X,Y)$ is a tiling of $G$ if every element of $G$ can be uniquely expressed as $xy$ for some $x\in X$ and $y\in Y$. In 1966, Rothaus and Thompson studied whether the symmetric…

Let $S_n$ denote the symmetric group of order $n$. Say that two subsets $x, y\subseteq S_n$ are \emph{equivalent} if there exist permutations $g_1, g_2\in S_n$ such that $g_1xg_2=y$, where multiplication is understood elementwise. Recently,…

This paper aims to address the relation between a magic square of odd order $n$ and a group, and their properties. By the modulo number $n$, we construct entries for each table from initial table of magic square with large number $n^2$.…

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any…

We introduce a new family of one-player games, involving the movement of coins from one configuration to another. Moves are restricted so that a coin can be placed only in a position that is adjacent to at least two other coins. The goal of…

An inglenook puzzle is a classic shunting (switching) puzzle often found on model railway layouts. A collection of wagons sits in a fan of sidings with a limited length headshunt (lead track). The aim of the puzzle is to rearrange the…