An overview of the fifteen puzzle using the ancient Chinese concept of pairing

The ancient Chinese mathematician Yang Hui was interested in magic squares and apparently constructed them by pair‐wise transpositions based on opposites by position—top‐bottom, left‐right, etc. The transpositions considered for solutions to the modern‐day fifteen puzzle, however, are much more restrictive; only adjacent horizontal and vertical moves are permitted. Nevertheless, we can conclude that only configurations which represent an even number of transpositions produce possible configurations in the fifteen puzzle whereas any configurations which represent an odd number of transpositions of the integers from their natural order are not possible in the fifteen puzzle.     

Topological insights from the Chinese Rings

L. Kauffman conjectured that a particular solution of the Chinese Rings puzzle is the simplest possible. We prove his conjecture by using low-dimensional topology and group theory. We notice also a surprising connection between the Chinese Rings and Habiro moves (related to Vassiliev invariants).

     

Two-player Tower of Hanoi

The Tower of Hanoi game is a classical puzzle in recreational mathematics (Lucas 1883) which also has a strong record in pure mathematics. In a borderland between these two areas we find the characterization of the minimal number of moves, which is \(2^n-1\), to transfer a tower of n disks. But there are also other variations to the game, involving for example real number weights on the moves of the disks. This gives rise to a similar type of problem, but where the final score seeks to be optimized. We study extensions of the one-player setting to two players, invoking classical winning conditions in combinatorial game theory such as the player who moves last wins, or the highest score wins. Here we solve both these winning conditions on three pegs.     

The Towers of Hanoi

This article presents an investigation carried out with a groupof able mathematics students who were studying at a level 1year in advance of their peers. The purpose was to investigatethe extension of usual three peg Towers of Hanoi to four pegsand attempt to find a rule that could be used to predict theminimum number of moves required to complete the puzzle.     

资本资产定价研究:基于损失厌恶、追赶时髦和习惯形成的新进展

为了推进资产定价理论的研究以及更好的解释重要的市场异象,本文通过在效用函数中引入投资者的异质假定,构造了一个投资者偏好由习惯形成,追赶时髦以及损失厌恶共同决定的效用函数,并在这个更加真实的效用函数的基础上建立了一个能够更好的解释市场异象的新的消费基础资本资产定价模型.另外,文章运用欧拉方程推出了模型的资产收益定价方程.通过定价方程,我们可以期望更好的同时解释溢价之谜,无风险利率之谜等重要的市场异象.这证明了引入合理的行为偏好才是解决股票溢价等问题的关键.     

Designs,Codes and Crypts—A Puzzle Altogether

In this paper we present an algebraic specificationof a puzzle commercially available on the toy market which isclosely related to the area of designs, codes and cryptography.The purpose of this article is to show the different ways ofapproaching the problem of solving this puzzle. An algebraicapproach for finding a solution is given. The seemingly unknowncombinatorial type of this puzzle and the lack of a general theoreticalbackground have stimulated the set-up of this paper to be presentedin this volume.     

益智游戏“汉诺塔”中的矩阵运算

汉诺塔(Tower of Hanoi)问题源于印度一个古老传说,据此做成了益智游戏,蕴含大量的数学思想与方法.本文采用矩阵描述汉诺塔状态和圆盘移动过程,将圆盘从一个位置移动到另一个位置转化为矩阵的加法,进而构造由若干可能状态矩阵组成的图的邻接矩阵,计算其幂矩阵,由此很方便地求得完成汉诺塔游戏的所有可能的圆盘移动方案,求解过程简单,含义清晰,易于理解和实现.     

Using the Tower of Hanoi puzzle to infuse your mathematics classroom with computer science concepts

This article suggests that logic puzzles, such as the well-known Tower of Hanoi puzzle, can be used to introduce computer science concepts to mathematics students of all ages. Mathematics teachers introduce their students to computer science concepts that are enacted spontaneously and subconsciously throughout the solution to the Tower of Hanoi puzzle. These concepts include, but are not limited to, conditionals, iteration, and recursion. Lessons, such as the one proposed in this article, are easily implementable in mathematics classrooms and extracurricular programmes as they are good candidates for ‘drop in’ lessons that do not need to fit into any particular place in the typical curriculum sequence. As an example for readers, the author describes how she used the puzzle in her own Number Sense and Logic course during the federally funded Upward Bound Math/Science summer programme for college-intending low-income high school students. The article explains each computer science term with real-life and mathematical examples, applies each term to the Tower of Hanoi puzzle solution, and describes how students connected the terms to their own solutions of the puzzle. It is timely and important to expose mathematics students to computer science concepts. Given the rate at which technology is currently advancing, and our increased dependence on technology in our daily lives, it has become more important than ever for children to be exposed to computer science. Yet, despite the importance of exposing today's children to computer science, many children are not given adequate opportunity to learn computer science in schools. In the United States, for example, most students finish high school without ever taking a computing course. Mathematics lessons, such as the one described in this article, can help to make computer science more accessible to students who may have otherwise had little opportunity to be introduced to these increasingly important concepts.     

Hanoi塔问题的一个公式解

Hanoi塔问题自提出以来已有一百多年的历史.其间,这一问题吸引了许多的研究者.正如H.A.Simon所指出的,Hanoi塔问题对于认知科学就象大肠杆菌对现代基因学那样,是一个无价的研究标本.事实上,它已成为组合数学,人工智能,计算机科学以及规划等中的递归问题的典型例子,并由此产生了各种各样成熟的算法.回顾这些结果,我们提出一个基本问题能否对Hanoi塔问题给出一个公式解?本文就此给出了一个肯定的回答.在我们的研究中,图论将是一个有力的工具     

Solving jigsaw puzzles by computer

An algorithm to assemble large jigsaw puzzles using curve matching and combinatorial optimization techniques is presented. The pieces are photographed one by one and then the assembly algorithm, which uses only the puzzle piece shape information, is applied. The algorithm was experimented successfully in the assembly of 104-piece puzzles with many almost similar pieces. It was also extended to solve an intermixed puzzle assembly problem and has successfully solved a 208-piece puzzle consisting of two intermixed 104-piece puzzles. Previous results solved puzzles with about 10 pieces, which were substantially different in shape.Work on this paper has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation.     

优化搜索原则创建数独模型

针对目前流行的数独游戏,建立数学模型,给出了创建不同难度的数独游戏的算法.首先,模型创造性地利用了数独游戏的解题算法的逆过程来创建一个完整的终盘.在此基础上,进一步优化创建终盘的搜索法则,保证了所采用的算法复杂度最小.最后,移去已有终盘上的一些格子以得到一道数独题的初盘,并在此过程中始终保证该问题有且只有唯一解.同时,基于人性化的原则,我们通过玩家使用的解题技巧和频率来划分难度.根据划分的难度等级,依次创建不同难度的数独游戏.     

卖空约束下的公司债券定价

信用溢价之谜是近年来在资产定价领域中热点研究问题之一,本文试图从结构化模型角度对这一问题做出解释。通过引入市场卖空约束条件,我们建立了新的公司债券定价模型。由于卖空约束在现实市场中普遍存在,因此建立这种结构化模型是具有实际意义的。实证研究表明该模型可以得出比现有模型更高的信用溢价,能很好地解释信用溢价之谜。     

On the classical descriptions of the quantum phenomena in the harmonic oscillator and in a charged particle under the coulomb force

In this work it is shown that the intrinsic phenomenon (the quantization of the energy) that appears in the first and simple systems studied initially by the quantum theory as the harmonic oscillator and the movement of a charged particle under the Coulomb force, can be obtained from the study of dissipative systems. In others words, we show that this phenomenon of the quantization of the energy of a particle which moves as an harmonic oscillator and which loses and wins energy can be obtained via a classical system of equations. The same also applies to the phenomena of the quantization of the energy of a charged particle which moves under the Coulomb force and which loses and wins energy.     

On the classical descriptions of the quantum phenomena in the harmonic oscillator and in a charged particle under the coulomb force

In this work it is shown that the intrinsic phenomenon (the quantization of the energy) that appears in the first and simple systems studied initially by the quantum theory as the harmonic oscillator and the movement of a charged particle under the Coulomb force, can be obtained from the study of dissipative systems. In others words, we show that this phenomenon of the quantization of the energy of a particle which moves as an harmonic oscillator and which loses and wins energy can be obtained via a classical system of equations. The same also applies to the phenomena of the quantization of the energy of a charged particle which moves under the Coulomb force and which loses and wins energy.     

The hapless frog and the evil wizard: an introduction to indirect proof

Students are introduced to indirect proof by a method of discovery.Initially, they are confronted with an intractable puzzle thatthey must either solve or expose as a hoax. Guided by timelyclues, they are unwittingly led to discredit the puzzle by applyingthe strategy of indirect proof.     

Generalizing the Disappearing Act: A Reply to István Aranyosi

In “The Reappearing Act” István Aranyosi postulates a new way of seeing to solve a puzzle posed in “The Disappearing Act;” an object that is exactly shaded can be seen simply by virtue of its contrast with its environment – just like a shadow. This object need not reflect, refract, absorb or block light. To undermine the motive for this heretical innovation, I generalize the puzzle to situations involving inexact shading. Aranyosi cannot extend his solution to these variations because he needs to conserve principles of camouflage. On the bright side, the solution to the puzzle that I propose in my book Seeing Dark Things does extend to these variations.     

The Reappearing Act

In his latest book, Roy Sorensen offers a solution to a puzzle he put forward in an earlier article -The Disappearing Act. The puzzle involves various question about how the causal theory perception is to be applied to the case of seeing shadows. Sorensen argues that the puzzle should be taken as bringing out a new way of seeing shadows. I point out a problem for Sorensen’s solution, and offer and defend an alternative view, according to which the puzzle is to be interpreted as showing a new way of seeing objects, in virtue of their contrast with light.

A repairing technique for the local search of the job-shop problem

The local search technique has become a widely used tool for solving many combinatorial optimization problems. In the case of the job-shop the implementation of such a technique is not straightforward at all due to the existence of the technological constraints among the operations that belong to the same job. Their presence renders a certain set of schedules infeasible. Consequently, special attention is required when defining optimization algorithms to prevent the possibility of reaching an infeasible schedule during execution. Traditionally, the problem is tackled on the neighborhood level by using only a limited set of moves for which feasibility inherently holds. This paper proposes an alternative way to avoid infeasibility by incorporating a repairing technique into the mechanism for applying moves to a schedule. Whenever an infeasible move is being applied, a repairing mechanism rearranges the underlying schedule in such a way that the feasibility of the move is restored. The possibility of reaching infeasible solutions is, therefore, eliminated on the lowest possible conceptual level. Consequently, neighborhood functions need not to be constrained to a limited set of feasible moves any more.