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The Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \times 6$. In both rou …

Consider the octagon in the Cartesian plane with vertices at (0,0), (1,0), (1,2), (4,2), (4,6), (7,2), (7,8), and (0,8). Are there other (infinitely many) polygons, such as this, lying entirely in the …

Is 47 the largest number which has a unique partition into five parts (15, 10, 10, 6, 6), no two of which are relatively prime?

The vertices of the Petersen graph (or any other simple graph) can be labelled in infinitely many ways with positive integers so that two vertices are joined by an edge if, and only if, the correspond …

Two players, Alice and Bob, take turns constructing a sequence $a_1,a_2,a_3,\dots$, of distinct positive integers, none greater than a given parameter $K$. Alice plays first and makes $a_1=1$. Thereaf …

For which positive integers N does there exist a square that can be completely tiled with N rectangles of integer sides whose areas or perimeters are precisely 1, 2, 3, ..., N?

Given a partition of an integer $N$, its $P$-graph is the graph whose vertices are its parts, two of which are joined by an edge if and only if they have a common divisor greater than one (i.e. they a …

Do integers n greater than 2 exist such that all the squares of sides 1, 2, 3, ..., n can be partitioned into two or more sets (none a singleton) each of whose squares can be used to tile a rectangle? …

Is anything known about the following? I hold in my hand a shuffled pack of cards numbered $1$ to $n$. One by one, I place them all, face up, on a table in piles. For each card I deal from my hand, sa …

An oblong is a rectangle whose width and length are consecutive integers: 1x2, 2x3, 3x4, etc. Does N exist such that it is possible to split the first N oblongs into 2 or more non-intersecting sets so …

Given a multiset of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if they have a common divisor greater than …

Freddy Barrera has shown that $k(1000)>5$ by verifying that every graph with fewer than 6 vertices (other than the singleton) is the P-graph of at least two partitions of 1000. On the other hand, from …

Given a set of positive integers, its common divisor graph ( CD-graph) is the graph whose vertices are the integers, two of which are joined by an edge if (and only if) they have a common divisor grea …

The vertices of any simple graph can be labeled in infinitely many ways with positive integers so that two vertices are joined by an edge if, and only if, they have a common divisor greater than 1. …

Is it possible to place the twelve pentominoes around a circle in such a way that if two of the pentominoes find themselves next to each other, it is because one of the two can be obtained from the ot …