Related papers: Conway's Wizards

We analyze a bead sliding game and relate it to some classical convex inequalities.

An open problem concerning Riemann sums, posed by O. Furdui, is considered.

In this expository note, we discuss a ``balls-and-urns'' probability puzzle posed by Daniel Litt.

In this article, we reveal how Benjamin Franklin constructed his second $8 \times 8$ magic square. We also construct two new $8 \times 8$ Franklin squares.

We give a new demonstration of Loewner's characterization of polynomials, solving in the positive a conjecture proposed by Laird and McCann in 1984.

We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a…

Several results about the union-closed sets conjecture are presented.

We deduce a recent theorem by R. Schwartz on the structure of the so-called Poncelet grid from complete integrability of the billiard in an ellipse

Collatz Conjecture sequences increase and decrease in seemingly random fashion. By identifying and analyzing the forms of numbers, we discover that Collatz sequences are governed by very specific, well-defined rules, which we call cascades.

Puzzles based on coloured cubes and other coloured geometrical figures have a long history in the recreational mathematical literature. One of the most commercially famous of these puzzles is the Instant Insanity that consists of four…

This paper describes a method used to construct infinitely many probable counterexamples of the abc conjecture over the rational integers.

We prove that the smallest elements of Shi parts and cone type parts exist and form Garside shadows. The latter resolves a conjecture of Parkinson and the second author as well as a conjecture of Hohlweg, Nadeau and Williams.

Riddles are concise linguistic puzzles that describe an object or idea through indirect, figurative, or playful clues. They are a longstanding form of creative expression, requiring the solver to interpret hints, recognize patterns, and…

We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again…

Peg solitaire is an old puzzle with a 300 year history. We consider two ways a computer can be utilized to find interesting peg solitaire puzzles. It is common for a peg solitaire puzzle to begin from a symmetric board position, we have…

Horn's conjecture, which given the spectra of two Hermitian matrices describes the possible spectra of the sum, was recently settled in the affirmative. In this survey we discuss one of the many steps in this, which required us to introduce…

Whenever a distinct state is immersed in a sea of complicated and dense states, the strength of the distinct state, which we refer to as a doorway, is distributed in their neighboring states. We analyze this mechanism for 2-D billiards with…

The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its "shortest" diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster…

The puzzle method was introduced by Choi and Park as an effective method for finding non-singular characteristic maps over wedged simplicial complexes $K(J)$ obtained from a given simplicial complex $K$. We study further the mod 2 case of…

In this work we resolve several conjectures stated in the On-Line Encyclopedia of Integer sequences.