Related papers: Picture-Hanging Puzzles
This paper covers a variety of mathematical folk puzzles, including geometric (Tangrams, dissection puzzles), logic, algebraic, probability (Monty Hall Problem, Birthday Paradox), and combinatorial challenges (Eight Queens Puzzle, Tower of…
We determine all composition-closed equational classes of Boolean functions. These classes provide a natural generalization of clones and iterative algebras: they are closed under composition, permutation and identification…
Single--photon which is initially uncorrelated with atom, will evolve to be entangled with the atom on their continuous kinetic variables in the process of resonant scattering. We find the relations between the entanglement and their…
We construct an explicit Hamiltonian cycle in the state graph of the 5-puzzle on a toroidal 2x 3 grid, a graph with 720 vertices. The cycle is described by a short symbolic sequence of 48 moves over the alphabet {L,R,V}, repeated $15$…
We prove that two polygons $A$ and $B$ have a reversible hinged dissection (a chain hinged dissection that reverses inside and outside boundaries when folding between $A$ and $B$) if and only if $A$ and $B$ are two noncrossing nets of a…
A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure we consider a knot…
Random projections offer an appealing and flexible approach to a wide range of large-scale statistical problems. They are particularly useful in high-dimensional settings, where we have many covariates recorded for each observation. In…
Boolean networks are special types of finite state time-discrete dynamical systems. A Boolean network can be described by a function from an n-dimensional vector space over the field of two elements to itself. A fundamental problem in…
It is well-known that there exist bar-joint frameworks (without continuous flexions) whose physical models can snap between different realizations due to non-destructive elastic deformations of material. We present a method to measure these…
Let $k\geq 2$ and fix a $k$-uniform hypergraph $\mathcal{F}$. Consider the random process that, starting from a $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices, repeatedly deletes the edges of a copy of $\mathcal{F}$ chosen uniformly…
A popular problem asks for the equilibrium separation between two identical (mutually repelling) charges suspended by strings fastened to a common point. We slightly modify this problem by considering two opposite (mutually attracting)…
A random jigsaw puzzle is constructed by arranging $n^2$ square pieces into an $n \times n$ grid and assigning to each edge of a piece one of $q$ available colours uniformly at random, with the restriction that touching edges receive the…
We study several bialgebraic structures on boolean functions, that is to say maps defined on the set of subsets of a finite set $X$, taking the value $0$ on $\emptyset$. Examples of boolean functions are given by the indicator function of…
A puzzle about prisoners trying to identify the color of a hat on their head leads to a version where there are k more hats than prisoners. This generalized puzzle is related to the independence number of the arrangement graph A(m, n) and…
A freely falling chain from a cup at certain height can jump. The process can be divided into two parts: a stable suspension and an accelerating procedure. Variational principle and force analysis demonstrate that the shape of stable…
You can invent striking and challenging problems with unique solution by building some symmetry into functional equations. Some are suitable for high school; others could generate college-level projects involving computer algebra. The…
We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this…
Structures which appear to move at or near the velocity of light contain singular points. Energy generated by the motion accumulates at these points into ever-narrowing peaks. In this paper we show that energy is generated by a curious…
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…
A long-standing quantum-mechanical puzzle is whether the collapse of the wave function is a real physical process or simply an epiphenomenon. This puzzle lies at the heart of the measurement problem. One way to choose between the…