Related papers: Conway's napkin problem
The Sleeping Beauty problem is a probability riddle with no definite solution for more than two decades and its solution is of great interest in many fields of knowledge. There are two main competing solutions to the problem: the halfer…
The "double Dixie cup problem" of D.J. Newman and L. Shepp (1960) is a well-known variant of the coupon collector's problem, where the object of study is the number $T_{m}(N)$ of coupons that a collector has to buy in order to complete $m$…
Many years ago John Tyrell a lecturer at King's college London challenged his Ph.D. students with the following puzzle: show that there is a unique triangle of minimal perimeter with exactly one vertex to lie on one of three given lines,…
The bunkbed conjecture was first posed by Kasteleyn. If $G=(V,E)$ is a finite graph and $H$ some subset of $V$, then the bunkbed of the pair $(G,H)$ is the graph $G\times\{1,2\}$ plus $|H|$ extra edges to connect for every $v\in H$ the…
We study a labeled variant of the classical Coupon Collector Problem (CCP), recently introduced by Tan et al., where coupons arrive in groups and only the set of labels is revealed. The goal is to determine the expected number of group…
In 1967, Klarner proposed a problem concerning the existence of reflecting $n$-queens configurations. The problem considers the feasibility of placing $n$ mutually non-attacking queens on the reflecting chessboard, an $n\times n$ chessboard…
Consider $2k-1$ voters, each of which has a preference ranking between $n$ given alternatives. An alternative $A$ is called a Condorcet winner, if it wins against every other alternative $B$ in majority voting (meaning that for every other…
The Sleeping Beauty problem is a puzzle in probability theory that has gained much attention since Elga's discussion of it [Elga, Adam, Analysis 60 (2), p.143-147 (2000)]. Sleeping Beauty is put asleep, and a coin is tossed. If the outcome…
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…
Sprouts is a two-player topological game, invented in 1967 in the University of Cambridge by John Conway and Michael Paterson. The game starts with p spots, and ends in at most 3p-1 moves. The first player who cannot play loses. The…
Conway's 99-graph problem is the second problem amongst the five 1000\$ 2017 open problems set. Four out of the five remain unsolved to this day, including the 99-graph problem. In this paper we quote Conway's definition of the problem and…
In this note, we introduce a distributed twist on the classic coupon collector problem: a set of $m$ collectors wish to each obtain a set of $n$ coupons; for this, they can each sample coupons uniformly at random, but can also meet in…
We generalize the well-known Coupon Collector Problem (CCP) in combinatorics. Our problem is to find the minimum and expected number of draws, with replacement, required to recover $n$ distinctly labeled coupons, with each draw consisting…
We consider Lionel Levine's notorious hat puzzle with two players. Each player has a stack of hats on their head, and each hat is chosen independently to be either black or white. After observing only the other player's hats, players…
A famous (and hard) chess problem asks what is the maximum number of safe squares possible in placing $n$ queens on an $n\times n$ board. We examine related problems from placing $n$ rooks. We prove that as $n\to\infty$, the probability…
We study in this paper a generalized coupon collector problem, which consists in determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an…
Let us call a sequence of numbers heapable if they can be sequentially inserted to form a binary tree with the heap property, where each insertion subsequent to the first occurs at a leaf of the tree, i.e. below a previously placed number.…
We study the existence of fair distributions when we have more guests than pieces to allocate, focusing on envy-free distributions among those who receive a piece. The conditions on the demand from the guests can be weakened from those of…
The Sleeping Beauty Problem remains a paradoxical problem that penetrates multiple disciplines that include probability theory, self-locating belief, decision theory, cognitive science, the philosophy of mathematics and science. It asks the…
We study popular matchings in three classical settings: the house allocation problem, the marriage problem, and the roommates problem. In the popular matching problem, (a subset of) the vertices in a graph have preference orderings over…