相关论文: Inverting the 10-Coin Triangle Puzzle and Other Sh…

We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length…

Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two…

A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of…

The 15 puzzle is a classic reconfiguration puzzle with fifteen uniquely labeled unit squares within a $4 \times 4$ board in which the goal is to slide the squares (without ever overlapping) into a target configuration. By generalizing the…

Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors,…

We present here necessary and sufficient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse. The techniques we use are related with the…

In Descartes' five circle problem integer curvatures (inverse radii) are considered. The positive integer curvature triple [c_1, c_2, c_3] (dimensionless), with non-decreasing entries for three given mutually touching circles, leading to…

There are thousands of papers on rootfinding for nonlinear scalar equations. Here is one more, to talk about an apparently new method, which I call ``Inverse Cubic Iteration'' (ICI) in analogy to the Inverse Quadratic Iteration in Richard…

A flipturn is an operation that transforms a nonconvex simple polygon into another simple polygon, by rotating a concavity 180 degrees around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of…

We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted…

Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the…

Recall the classical 15-puzzle, consisting of 15 sliding blocks in a $4\times 4$ grid. Famously, the configuration space of this puzzle consists of two connected components, corresponding to the odd and even permutations of the symmetric…

We present an analysis of a coin-tossing problem posed by Daniel Litt which has generated some popular interest. We demonstrate a recursive identity which leads to relatively simple formulas for the excess number of wins for one player over…

In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…

If $A$ is a tridiagonal matrix, then the equations $AX=I$ and $XA=I$ defining the inverse $X$ of $A$ are in fact the second order recurrence relations for the elements in each row and column of $X$. Thus, the recursive algorithms should be…

We investigate the reconfiguration of $n$ blocks, or "tokens", in the square grid using "line pushes". A line push is performed from one of the four cardinal directions and pushes all tokens that are maximum in that direction to the…

Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new "focus-inversive" family inscribed in Pascal's Lima\c{c}on. The following are some of its surprising invariants: (i)…

We revisit the game in which each of several players chooses a pattern and then a coin is flipped repeatedly until one of these patterns is generated. In particular, we demonstrate how to compute the probability of any one player winning…

We investigate a type of a Sudoku variant called Sudo-Kurve, which allows bent rows and columns, and develop a new, yet equivalent, variant we call a Sudo-Cube. We examine the total number of distinct solution grids for this type with or…

Multiplicative inverse is a crucial operation in public key cryptography, and been widely used in cryptography. Public key cryptography has given rise to such a need, in which we need to generate a related public and private pair of…