相关论文: Computer-aided solution to the $k$-bonacci pick-up…

Consider the following process whereby $n$ balls are distributed into $k$ bins. Repeatedly, a ball is removed from a non-empty bin chosen uniformly at random. The process ends when a single non-empty bin remains. Will Ma…

We apply a classic result in Theoretical Computer Science to solve a sticky problem, and give a neat and slick quick proof of half of it

This paper questions the generally accepted assumption that one can make a random choice that is independent of the rest of the universe. We give a general description of any setup that could be conceived to generate random numbers. Based…

In this paper we enumerate the number of ways of selecting $k$ objects from $n$ objects arrayed in a line such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects and provide three different formulas when $m,p\geq 1$ and…

We introduce the problem of finding a set $B$ of $k$ points in $[0,1]^n$ such that the expected cost of the cheapest point in $B$ that dominates a random point from $[0,1]^n$ is minimized. We study the case where the coordinates of the…

Let $C_n^k$ be the $k$-th power of a cycle on $n$ vertices (i.e. the vertices of $C_n^k$ are those of the $n$-cycle, and two vertices are connected by an edge if their distance along the cycle is at most $k$). For each vertex draw uniformly…

We construct irreducible balanced non-transitive sets of $n$-sided dice for any positive integer $n$, which was raised in \cite[Question 5.2]{SS17}. One main tool of the construction is to study so-called fair sets of dice. Furthermore, we…

The problems of Hadamard quantum coin flipping in n-trials and related generalized Fibonacci sequences of numbers were introduced in [1]. It was shown that for an arbitrary number of repeated consecutive states, probabilities are determined…

Given a set $P$ of $n$ points in $\mathbf{R}^d$, and a positive integer $k \leq n$, the $k$-dispersion problem is that of selecting $k$ of the given points so that the minimum inter-point distance among them is maximized (under Euclidean…

As a generalization of polyominoes we consider edge-to-edge connected nonoverlapping unions of regular $k$-gons. For $n\le 4$ we determine formulas for the number $a_k(n)$ of generalized polyominoes consisting of $n$ regular $k$-gons.…

In this work, the probability of return for random walks on $\mathbb{Z}$, whose increment is given by the $k$-bonacci sequence, is determined. Also, the Hausorff, packing and box-counting dimensions of the set of these walks that return an…

An equilateral stick number $s_{=}(K)$ of a knot $K$ is defined to be the minimal number of sticks required to construct a polygonal knot of $K$ which consists of equal length sticks. Rawdon and Scharein [12] found upper bounds for the…

Consider a set $P$ of $n$ points picked uniformly and independently from $[0,1]^d$ for a constant dimension $d$ -- such a point set is extremely well behaved in many aspects. For example, for a fixed $r \in [0,1]$, we prove a new…

Inspired by the Japanese game Pachinko, we study simple (perfectly "inelastic" collisions) dynamics of a unit ball falling amidst point obstacles (pins) in the plane. A classic example is that a checkerboard grid of pins produces the…

A variant of the classical knapsack problem is considered in which each item is associated with an integer weight and a qualitative level. We define a dominance relation over the feasible subsets of the given item set and show that this…

How many fair coin tosses to choose 1 of $n$ options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We…

Let $\mathbb{P}_{\kappa}(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_\kappa$, a regular $\kappa$-gon with area $1$, are in convex position, that is, form the vertex set of a…

Computing sets of high quality solutions has gained increasing interest in recent years. In this paper, we investigate how to obtain sets of optimal solutions for the classical knapsack problem. We present an algorithm to count exactly the…

We study in this paper a generalized coupon collector problem, which consists in analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We…

We define an $n$-gon to be any convex polygon with $n$ vertices. Let $V$ represent the set of vertices of the polygon. A proper $k$-coloring refers to a function, $f$ : $V$ $\rightarrow$ $\{1, 2, 3, ... k\}$, such that for any two vertices…