Abstract In this paper a new method for solving the problem of placing n queens on a n×n chessboard such that no two queens directly threaten one another and considering that several immovable queens are already occupying established positions on the board is presented. At first, it is applied to the 8–Queens puzzle on a classical chessboard and finally to the n Queens completion puzzle. Furthermore, this method allows finding repetitive patterns of solutions for any n.
{"title":"An alternative algorithm for the n–Queens puzzle","authors":"David Luque Sacaluga","doi":"10.2478/rmm-2021-0003","DOIUrl":"https://doi.org/10.2478/rmm-2021-0003","url":null,"abstract":"Abstract In this paper a new method for solving the problem of placing n queens on a n×n chessboard such that no two queens directly threaten one another and considering that several immovable queens are already occupying established positions on the board is presented. At first, it is applied to the 8–Queens puzzle on a classical chessboard and finally to the n Queens completion puzzle. Furthermore, this method allows finding repetitive patterns of solutions for any n.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124989252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Experimentation, gathering data, and computation are an integral part of the process whereby mathematicians discover theorems and their proofs. Mathematics may be a science of exact proof, but the data and the process used in the discovery of certain proofs adds an experimental component. And, computers can help by creating data that reveal patterns. This productive methodology is one worth emphasizing at every level of mathematical instruction, but especially in primary, middle and high school curricula, where the emphasis on algorithms, methods, technique, and vocabulary leaves experiment and discovery as an afterthought. Inspired by a question posted by Marc Dostie on the Rediscovering Mathematics Facebook page 1, we consider a number of problems related to the following door lock. We offer this exploration as an example of how to incorporate discovery, experiment, and calculation into mathematics and pedagogy. An early version of this work was presented in an invited lecture at the Mathematics and Computer Science Colloquium at Providence College in 2015.2 Programmable door locks such as the one in the figure, commonly found in schools, hospitals, and office buildings, provide a flexible way to maintain selective security and entry to different rooms and areas of buildings. To enter a room, a person presses certain buttons, then enter, and turns the handle. In this particular model of the lock, once a button is pressed, it cannot be pressed again, however, buttons can be pressed simultaneously, and the order in which the presses occur is significant.
{"title":"The Five-Button Door Lock – Experiment and Discovery in Mathematics","authors":"S. Simonson, T. Woodcock","doi":"10.2478/rmm-2021-0006","DOIUrl":"https://doi.org/10.2478/rmm-2021-0006","url":null,"abstract":"Experimentation, gathering data, and computation are an integral part of the process whereby mathematicians discover theorems and their proofs. Mathematics may be a science of exact proof, but the data and the process used in the discovery of certain proofs adds an experimental component. And, computers can help by creating data that reveal patterns. This productive methodology is one worth emphasizing at every level of mathematical instruction, but especially in primary, middle and high school curricula, where the emphasis on algorithms, methods, technique, and vocabulary leaves experiment and discovery as an afterthought. Inspired by a question posted by Marc Dostie on the Rediscovering Mathematics Facebook page 1, we consider a number of problems related to the following door lock. We offer this exploration as an example of how to incorporate discovery, experiment, and calculation into mathematics and pedagogy. An early version of this work was presented in an invited lecture at the Mathematics and Computer Science Colloquium at Providence College in 2015.2 Programmable door locks such as the one in the figure, commonly found in schools, hospitals, and office buildings, provide a flexible way to maintain selective security and entry to different rooms and areas of buildings. To enter a room, a person presses certain buttons, then enter, and turns the handle. In this particular model of the lock, once a button is pressed, it cannot be pressed again, however, buttons can be pressed simultaneously, and the order in which the presses occur is significant.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"166 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133225289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mi-Hee Han, Ella Kim, Evin Liang, Miriam Mira Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui, Cindy Wei, T. Khovanova
Abstract Do you want to know what an anti-chiece Latin square is? Or what a non-consecutive toroidal modular Latin square is? We invented a ton of new types of Latin squares, some inspired by existing Sudoku variations. We can’t wait to introduce them to you and answer important questions, such as: do they even exist? If so, under what conditions? What are some of their interesting properties? And how do we generate them?
{"title":"Fun with Latin Squares","authors":"Mi-Hee Han, Ella Kim, Evin Liang, Miriam Mira Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui, Cindy Wei, T. Khovanova","doi":"10.2478/rmm-2023-0003","DOIUrl":"https://doi.org/10.2478/rmm-2023-0003","url":null,"abstract":"Abstract Do you want to know what an anti-chiece Latin square is? Or what a non-consecutive toroidal modular Latin square is? We invented a ton of new types of Latin squares, some inspired by existing Sudoku variations. We can’t wait to introduce them to you and answer important questions, such as: do they even exist? If so, under what conditions? What are some of their interesting properties? And how do we generate them?","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126802072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Dots-and-Boxes is a popular children’s game whose winning strategies have been studied by Berlekamp, Conway, Guy, and others. In this article we consider two variations, Dots-and-Triangles and Dots-and-Polygons, both of which utilize the same lattice game board structure as Dots-and-Boxes. The nature of these variations along with this lattice structure lends itself to applying Pick’s theorem to calculate claimed area. Several strategies similar to those studied in Dots-and-Boxes are used to analyze these new variations.
{"title":"Dots-and-Polygons","authors":"Jessica L. Dickson, R. Perrier","doi":"10.2478/rmm-2022-0002","DOIUrl":"https://doi.org/10.2478/rmm-2022-0002","url":null,"abstract":"Abstract Dots-and-Boxes is a popular children’s game whose winning strategies have been studied by Berlekamp, Conway, Guy, and others. In this article we consider two variations, Dots-and-Triangles and Dots-and-Polygons, both of which utilize the same lattice game board structure as Dots-and-Boxes. The nature of these variations along with this lattice structure lends itself to applying Pick’s theorem to calculate claimed area. Several strategies similar to those studied in Dots-and-Boxes are used to analyze these new variations.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128694139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A commonly occurring task in intelligence tests or recreational riddles is to “find the odd one out”, that is, to determine a unique element of a set of objects that is somehow special. It is somewhat arbitrary what exactly the relevant feature is that makes one object different. But once that is settled, the answer becomes obvious. Not so with a puzzle popularized by Tanya Khovanova to express her dislike for this type of puzzle. Here, it is a more complicated relation between the objects and the features that determines the odd object, because there is only one object that does not have a unique feature expression. This puzzle inspired me to look for even more complicated relations between objects, features and feature expressions that appear to be even more symmetric, but actually still single out a “special object”. This paper provides useful definitions, a theoretical basis, solution algorithms, and several examples for this kind of puzzle.
{"title":"Next Level Odd-One-Out Puzzles","authors":"Benjamin Berger","doi":"10.2478/rmm-2020-0003","DOIUrl":"https://doi.org/10.2478/rmm-2020-0003","url":null,"abstract":"\u0000 A commonly occurring task in intelligence tests or recreational riddles is to “find the odd one out”, that is, to determine a unique element of a set of objects that is somehow special. It is somewhat arbitrary what exactly the relevant feature is that makes one object different. But once that is settled, the answer becomes obvious. Not so with a puzzle popularized by Tanya Khovanova to express her dislike for this type of puzzle. Here, it is a more complicated relation between the objects and the features that determines the odd object, because there is only one object that does not have a unique feature expression. This puzzle inspired me to look for even more complicated relations between objects, features and feature expressions that appear to be even more symmetric, but actually still single out a “special object”. This paper provides useful definitions, a theoretical basis, solution algorithms, and several examples for this kind of puzzle.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132574826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Generala is a social multi-player dice game popular in Spanish and Portuguese speaking countries. Generala is similar to Yacht, Yatzy, Kniffel, and other close varieties. The most well known variant, a commercial game marketed as Yahtzee, still sells 50 million games each year. All five variants above use five dice and the player scores by matching the numbers on the dice with certain combinations. Generala has 10 such combinations: Ones, Twos, Threes, Fours, Fives, Sixes, Escalera (1-2-3-4-5 or 2-3-4-5-6), Full-House (XXXYY), Fourof-a-Kind (XXXXY), and Generala (XXXXX). In a turn, the player rolls all five dice and then may either stand or re-roll some of the dice. She may then re-roll some of the dice a second time, but after the third roll the player must score the five dice by matching them with one unused combination. Generala scores 50 points, Four-of-a-Kind 40, Full House 30 and Escalera 20. The Ones to Sixes score according to the number of dice with that value. For example, 1-1-3-4-4 scores 2 points as Ones, 3 points as Threes and 8 points as Fours. In some versions a player scores 5 or 10 bonus points when hitting a combination on the first roll, but here I will consider only the “Vanilla” version. The player may waive a combination, i.e. pick a combination which does not match the dice, for a score of zero. For example, a player might waive Generala with 1-11-1-5 if Ones is no longer available to score. Each combination can be scored exactly once. After completing one turn the dice pass to the player on the left, and this process continues until all players have completed 10 turns and either scored or waived all their combinations. The player with most points wins. As is true for most games, the origin and early history of Generala is mostly unknown, and it is simply considered a “traditional game”. Dice of various kinds pre-date recorded history and feature in 5, 000-year-old games such as Senet and the Royal Game of Ur
{"title":"Stay in Command: Optimal Play for Two Person Generala","authors":"Joseph Heled","doi":"10.2478/rmm-2020-0004","DOIUrl":"https://doi.org/10.2478/rmm-2020-0004","url":null,"abstract":"Generala is a social multi-player dice game popular in Spanish and Portuguese speaking countries. Generala is similar to Yacht, Yatzy, Kniffel, and other close varieties. The most well known variant, a commercial game marketed as Yahtzee, still sells 50 million games each year. All five variants above use five dice and the player scores by matching the numbers on the dice with certain combinations. Generala has 10 such combinations: Ones, Twos, Threes, Fours, Fives, Sixes, Escalera (1-2-3-4-5 or 2-3-4-5-6), Full-House (XXXYY), Fourof-a-Kind (XXXXY), and Generala (XXXXX). In a turn, the player rolls all five dice and then may either stand or re-roll some of the dice. She may then re-roll some of the dice a second time, but after the third roll the player must score the five dice by matching them with one unused combination. Generala scores 50 points, Four-of-a-Kind 40, Full House 30 and Escalera 20. The Ones to Sixes score according to the number of dice with that value. For example, 1-1-3-4-4 scores 2 points as Ones, 3 points as Threes and 8 points as Fours. In some versions a player scores 5 or 10 bonus points when hitting a combination on the first roll, but here I will consider only the “Vanilla” version. The player may waive a combination, i.e. pick a combination which does not match the dice, for a score of zero. For example, a player might waive Generala with 1-11-1-5 if Ones is no longer available to score. Each combination can be scored exactly once. After completing one turn the dice pass to the player on the left, and this process continues until all players have completed 10 turns and either scored or waived all their combinations. The player with most points wins. As is true for most games, the origin and early history of Generala is mostly unknown, and it is simply considered a “traditional game”. Dice of various kinds pre-date recorded history and feature in 5, 000-year-old games such as Senet and the Royal Game of Ur","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128495746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Although in a sequence of coin flips, any given consecutive set of, say, three flips is equally likely to be one of the eight possible, i.e., HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT, it is rather peculiar that one sequence of three is not necessarily equally likely to appear first as another set of three. This fact can be illustrated by the following game: you and your opponent each ante a penny. Each selects a pattern of three, and the umpire tosses a coin until one of the two patterns appears, awarding the antes to the player who chose that pattern. Your opponent picks HHH; you pick HTH. The odds, you will find, are in your favor. By how much?
{"title":"Making the Unfair Fair","authors":"R. Vallin","doi":"10.2478/rmm-2020-0002","DOIUrl":"https://doi.org/10.2478/rmm-2020-0002","url":null,"abstract":"Although in a sequence of coin flips, any given consecutive set of, say, three flips is equally likely to be one of the eight possible, i.e., HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT, it is rather peculiar that one sequence of three is not necessarily equally likely to appear first as another set of three. This fact can be illustrated by the following game: you and your opponent each ante a penny. Each selects a pattern of three, and the umpire tosses a coin until one of the two patterns appears, awarding the antes to the player who chose that pattern. Your opponent picks HHH; you pick HTH. The odds, you will find, are in your favor. By how much?","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130662912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider different ways to count the number of clues in American-style crossword puzzle grids. One yields a basic parity result for symmetric square grids. Another works efficiently even for non-symmetric grids that are already numbered. We further discuss the upper limit on the number of clues in a crossword puzzle with no 2-letter answers, and open questions are given. As a bonus, a mathematically-themed crossword puzzle is included!
{"title":"Counting Clues in Crosswords","authors":"Kevin Ferland","doi":"10.2478/rmm-2020-0001","DOIUrl":"https://doi.org/10.2478/rmm-2020-0001","url":null,"abstract":"\u0000 We consider different ways to count the number of clues in American-style crossword puzzle grids. One yields a basic parity result for symmetric square grids. Another works efficiently even for non-symmetric grids that are already numbered. We further discuss the upper limit on the number of clues in a crossword puzzle with no 2-letter answers, and open questions are given. As a bonus, a mathematically-themed crossword puzzle is included!","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"23 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114031113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract 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 user may run one algorithm at a time for the specified cost, or give up and pay the penalty. The probability of success may be implied by randomization in the algorithm, or by assuming a probability distribution on the input space, which lead to different variants of the problem. The goal is to minimize the expected cost of the process under the assumption that the algorithms are independent. We study several variants of this problem, and present possible solution strategies and a hardness result.
{"title":"Adjustable Coins","authors":"S. Moran, I. Yavneh","doi":"10.2478/rmm-2021-0009","DOIUrl":"https://doi.org/10.2478/rmm-2021-0009","url":null,"abstract":"Abstract 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 user may run one algorithm at a time for the specified cost, or give up and pay the penalty. The probability of success may be implied by randomization in the algorithm, or by assuming a probability distribution on the input space, which lead to different variants of the problem. The goal is to minimize the expected cost of the process under the assumption that the algorithms are independent. We study several variants of this problem, and present possible solution strategies and a hardness result.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130131101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Bishop Independence concerns determining the maximum number of bishops that can be placed on a board such that no bishop can attack any other bishop. This paper presents the solution to the bishop independence problem, determining the bishop independence number, for all sizes of boards on the surface of a square prism.
{"title":"Bishop Independence on the Surface of a Square Prism","authors":"L. H. Harris, S. Perkins, P. Roach","doi":"10.2478/rmm-2021-0007","DOIUrl":"https://doi.org/10.2478/rmm-2021-0007","url":null,"abstract":"Abstract Bishop Independence concerns determining the maximum number of bishops that can be placed on a board such that no bishop can attack any other bishop. This paper presents the solution to the bishop independence problem, determining the bishop independence number, for all sizes of boards on the surface of a square prism.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123508432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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