Related papers: How big a table do you need for your jigsaw puzzle…
In this work, we define a triangle area number to be the area number of a triangle whose sides have integer lengths, and whose area is a rational number. In Result 3, on page 17, we prove that every triangle area number is in fact an…
We consider the inverse problem of determining the possible presence of an inclusion in a thin plate by boundary measurements. The plate is made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. The…
Magic squares have been an enthralling topic in mathematics for centuries. They are formed by filling in all the cells of a square matrix with the numbers starting from one so that the sum of all rows, columns, and diagonals is the same.…
A sliding puzzle is a combination puzzle where a player slide pieces along certain routes on a board to reach a certain end-configuration. In this paper, we propose a novel measurement of complexity of massive sliding puzzles with…
Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$).…
We define a magic square to be a square matrix whose entries are nonnegative integers and whose rows, columns, and main diagonals sum up to the same number. We prove structural results for the number of such squares as a function of the…
Site swap is a mathematical notation used by jugglers to communicate, create and study complex juggling patterns. Determining the number of possible site swap juggling patterns with respect to certain limiting parameters such as number of…
We present a general algorithm for finding the overlap area between two ellipses. The algorithm is based on finding a segment area (the area between an ellipse and a secant line) given two points on the ellipse. The Gauss-Green formula is…
We estimate from below the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space and in a given linear order in the Euclidean 3-space. Similarly, we estimate the number…
In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the…
An associative magic square is a magic square such that the sum of any 2 cells at symmetric positions with respect to the center is constant. The total number of associative magic squares of order 7 is enormous, and thus, it is not…
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $\delta=\frac{8}{5\pi}\approx 0.509$. This implies that any set of (not…
We study a hierarchical model of non-overlapping cubes of sidelengths $2^j$, $j \in \mathbb{Z}$. The model allows for cubes of arbitrarily small size and the activities need not be translationally invariant. It can also be recast as a spin…
Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and…
The number of complete tilings of m X n floors for tiles of shape 1 X 2, 1 X 3, 1 X 4 and 2 X 3 is computed numerically for floors up to width m=9 and variable floor lengths n. Counts are obtained for two classes, for fixed tile stack…
We discuss the problem of counting certain LEGO structures, primarily those comprising parallel $w \times 1$ tiles. These can be combined, as a single LEGO structure, by interlocking the tiles. %Alternatively, if the interlocking condition…
What is the shape of the 2D convex region P from which, when 2 mutually congruent convex pieces with maximum possible area are cut out, the highest fraction of the area of P is left over? When P is restricted to the set of all possible…
Satisfying spin-assignments in triangulations of a surface are states of minimum energy of the antiferromagnetic Ising model on triangulations which correspond (via geometric duality) to perfect matchings in cubic bridgeless graphs. In this…
This paper deals with themes such as approximate counting/evaluation of the total number of flat-foldings for random origami diagrams, evaluation of the values averaged over various instances, obtaining forcing sets for general origami…
We study the dissection of a square into congruent convex polygons. Yuan \emph{et al.} [Dissecting the square into five congruent parts, Discrete Math. \textbf{339} (2016) 288-298] asked whether, if the number of tiles is a prime number…