相关论文: Baron Munchhausen's Sequence

An unbiased coin is tossed $n$ times independently and sequentially. In this paper, we will study the length of the longest consecutive switches, and prove that the limit behaviors are similar to that of the length of the longest head-run.

Since the introduction of the Kolmogorov complexity of binary sequences in the 1960s, there have been significant advancements in the topic of complexity measures for randomness assessment, which are of fundamental importance in theoretical…

We obtain partial progress towards answering the question of whether the quantity defined in the Mondrian Puzzle can ever equal 0. More specifically, we obtain a nontrivial lower bound for the cardinality of the set $\{n\leq x: M(n)\neq0…

We study the famous mathematical puzzle of prisoners and hats. We introduce a framework in which various variants of the problem can be formalized. We examine three particular versions of the problem (each one in fact a class of problems)…

We derive weighted sums, including binomial and double binomial sums, for the generalized Fibonacci sequence $\{G_m\}$ where for $m\ge 2$, $G_m=G_{m-1}+G_{m-2}$ with initial values $G_0$ and $G_1$.

We introduce an elementary congruence-based procedure to look for q-th power multiples in arbitrary binary recurrence sequences (q>2). The procedure allows to prove that no such multiples exist in many instances.

We study the Min-Weighted Sum Bin Packing problem, a variant of the classical Bin Packing problem in which items have a weight, and each item induces a cost equal to its weight multiplied by the index of the bin in which it is packed. This…

This paper presents a systematic method to solve difficult 9 x 9 Sudoku puzzles by hand. While computer algorithms exist to solve these puzzles, these algorithms are not good for human's to use because they involve too many steps and…

Mochon's proof [Moc07] of existence of quantum weak coin flipping with arbitrarily small bias is a fundamental result in quantum cryptography, but at the same time one of the least understood. Though used several times as a black box in…

The distribution of a given sequence in the set of all sequences with n ones and m = M - n zeros are found by relating the problem to the partitions of a natural number in m natural summands, taking into account the order. The formulas…

This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non-overlapping circles. The first half of the article is an exposition of…

The Monty Hal problem is an attractive puzzle. It combines simple statement with answers that seem surprising to most audiences. The problem was thoroughly solved over two decades ago. Yet, more recent discussions indicate that the solution…

The Chinos game is a non-cooperative game between players who try to guess the total sum of coins drawn collectively. Semiclassical and quantum versions of this game were proposed by F. Guinea and M. A. Martin-Delgado, in J. Phys. A: Math.…

How can we predict the difficulty of a Sudoku puzzle? We give an overview of difficulty rating metrics and evaluate them on extensive dataset on human problem solving (more then 1700 Sudoku puzzles, hundreds of solvers). The best results…

We extend the classical coupon collector's problem to one in which two collectors are simultaneously and independently seeking collections of $d$ coupons. We find, in finite terms, the probability that the two collectors finish at the same…

The locker puzzle is a game played by multiple players against a referee. It has been previously shown that the best strategy that exists cannot succeed with probability greater than 1-ln2 \approx 0.31, no matter how many players are…

We consider a somehow peculiar Token/Bucket problem which at first sight looks confusing and difficult to solve. The winning approach to solve the problem consists in going back to the simple and traditional methods to solve computer…

Contemporary machine learning methods will try to approach the Bayes error, as it is the lowest possible error any model can achieve. This paper postulates that any decision is composed of not one but two Bayesian decisions and that…

A linear algorithm is described for solving the n-Queens Completion problem for an arbitrary composition of k queens, consistently distributed on a chessboard of size n x n. Two important rules are used in the algorithm: a) the rule of…

The primary objective of this note is to revisit the two envelope problem and propose a simple resolution. It is argued that the paradox arises from the ambiguity associated with the money content $x of the chosen envelope. When X=x is…