Related papers: Finding Golden Nuggets by Reduction
The $n$-th Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by binary strings with no two consecutive ones. We determine $\pi(\Gamma_n) = 2^n$ for $n \le 6$, so the pebbling number of $\Gamma_n$ equals that of the…
A reformulation of the Kelly Criterion is presented. Let $\mathfrak{G}$ be a generic stochastic Bernoulli binary game with outcomes $\mathscr{Z}(I)\in\lbrace -1,1\rbrace$ of N trials for $I=1...N$. The binomial probabilities are…
The Fibonacci sequence is a series of positive integers in which, starting from $0$ and $1$, every number is the sum of two previous numbers, and the limiting ratio of any two consecutive numbers of this sequence is called the golden ratio.…
Stochastic games with discounted payoff, introduced by Shapley, model adversarial interactions in stochastic environments where two players try to optimize a discounted sum of rewards. In this model, long-term weights are geometrically…
We give a bijection between permutations of length 2n and certain pairs of Dyck paths with labels on the down steps. The bijection arises from a game in which two players alternate selecting from a set of 2n items: the permutation encodes…
The deduction game may be thought of as a variant on the classical game of cops and robber in which the cops (searchers) aim to capture an invisible robber (evader); each cop is allowed to move at most once, and cops situated on different…
Stochastic games are a classical model in game theory in which two opponents interact and the environment changes in response to the players' behavior. The central solution concepts for these games are the discounted values and the value,…
A combinatorial game is a two-player game without hidden information or chance elements. The disjunctive sum $G + H$ of games $G$ and $H$ is the game in which $G$ and $H$ are played in parallel, and a player makes a move on exactly one of…
Quantitative games are two-player zero-sum games played on directed weighted graphs. Total-payoff games (that can be seen as a refinement of the well-studied mean-payoff games) are the variant where the payoff of a play is computed as the…
Natural selection drives species to develop brains, with sizes that increase with the complexity of the tasks to be tackled. Our goal is to investigate the balance between the metabolic costs of larger brains compared to the advantage they…
The problem of the universal form of the size spectrum is analyzed. The half-widths of two wings of spectrum is introduced and it is shown that their ratio is very close to the golden fraction. In appendix it is shown that behind the golden…
We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer…
We introduce an evolutionary game with feedback between perception and reality, which we call the reality game. It is a game of chance in which the probabilities for different objective outcomes (e.g., heads or tails in a coin toss) depend…
When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set's underlying structure. We begin by investigating finite sets of…
We introduce a variant of Wythoff's Game that we call $m$-Modular Wythoff's Game. In the original Wythoff's Game, players can take a positive number of tokens from one pile, or they can take a positive number of tokens from both piles if…
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
This paper presents a generalization of the Kostant game, a combinatorial framework originally for generating positive roots in Lie algebras. By introducing an arbitrary multi-vertex modification, we prove that the resulting game…
We study algorithmic complexity of solving subtraction games in a~fixed dimension with a finite difference set. We prove that there exists a game in this class such that any algorithm solving the game runs in exponential time. Also we prove…
Given $n$ piles of tokens and a positive integer $k \leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and…
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…