Related papers: A note on the problem of prisoners and hats
We examine a new variant of the classic prisoners and lightswitches puzzle: A warden leads his $n$ prisoners in and out of $r$ rooms, one at a time, in some order, with each prisoner eventually visiting every room an arbitrarily large…
In this piece, we examine one variant of the infamous 15 Tile Puzzle and develop a mathematical backing behind why it is unsolvable. Using concepts of permutations, bijectivity, and cycle transpositions, we not only prove how to model this…
We investigate the complexity of a puzzle that turns out to be NL-complete.
The multi-armed bandit is a mathematical interpretation of the problem a gambler faces when confronted with a number of different machines (bandits). The gambler wants to explore different machines to discover which machine offers the best…
In this note some philosophical thoughts and observations about mathematics are expressed, arranged as challenges to some common claims.
We consider the problem of revealing a small hidden lattice from the knowledge of a low-rank sublattice modulo a given sufficiently large integer -- the {\em Hidden Lattice Problem}. A central motivation of study for this problem is the…
In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if…
In this article we prove the impossibility of some disentanglement puzzles, first building mathematical models that reflect the essential characteristics of these puzzles.
Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of $n$ players is assigned a hat from a given color set.…
Several sages wearing colored hats occupy the vertices of a graph. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbours without exchanging any information. Each hat can have one of…
Several different "hat games" have recently received a fair amount of attention. Typically, in a hat game, one or more players are required to correctly guess their hat colour when given some information about other players' hat colours.…
This paper is an overview and survey of work on the 3x+1 problem, also called the Collatz problem, and generalizations of it. It gives a history of the problem. It addresses two questions: (1) What can mathematics currently say about this…
This is a survey of old and new problems and results in additive number theory.
We study a cooperative game in which each member of a team of $N$ players, wearing coloured hats and situated at the vertices of a cycle graph $C_N$, is guessing their own hat colour merely on the basis of observing the hats worn by their…
In this paper, we bring a complete solution to the Ovals problem, as formulated in [3] and [24].
In this paper we consider a scenario where there are several algorithms for solving a given problem. Each algorithm is associated with a probability of success and a cost, and there is also a penalty for failing to solve the problem. The…
The quantum mechanical approach to the well known prisoners dilemma, one of the basic examples to illustrate the concepts of Game Theory, is implemented with a classical optical resource, nonquantum entanglement between spin and orbital…
This paper examines the integration of computational complexity into game theoretic models. The example focused on is the Prisoner's Dilemma, repeated for a finite length of time. We show that a minimal bound on the players' computational…
The purpose of this note is to raise two different questions, which are rarely if ever considered, and to which, it seems, we lack convincing, systematic answers. These questions can be posed as: - Why do we compute? - What do we compute?…
In this paper the standard prisoners' dilemma is embedded in environmental conditions in which the interaction takes place. This provides a theoretical background to the analysis of the empirical studies which indicate that including…