R. Ford, James Grime, Eric Harshbarger, Brian Pollock
Abstract Before a game begins, the players need to decide the order of play. This order of play is determined by each player rolling a die. Does there exist a set of dice such that draws are excluded and each order of play is equally likely? For four players the solution involves four 12-sided dice, sold commercially as Go First Dice. However, the solution for five players remained an open question. We present two solutions. The first solution has a particular mathematical structure known as binary dice, and results in a set of five 60-sided dice, where every place is equally likely. The second solution is an inductive construction that results in one one 36-sided die; two 48-sided dice; one 54-sided die; and one 20-sided die, where each permutation is equally likely.
在游戏开始前，玩家需要决定游戏的顺序。这个游戏顺序是由每个玩家掷骰子决定的。是否存在一组骰子，使得平局被排除在外，并且每种玩法顺序的可能性相等?对于四名玩家来说，解决方案涉及四个12面骰子，商业上以Go First dice的名义出售。然而，五名球员的解决方案仍然是一个悬而未决的问题。我们提出了两种解决方案。第一种解决方案有一种特殊的数学结构，称为二元骰子，结果是一组5个60面骰子，其中每个位置的概率都是相等的。第二个解决方案是归纳构造，结果是一个36面骰子;两个48面骰子;一个54面骰子;还有一个20面骰子，每种排列的概率都是一样的。
{"title":"Go First Dice for Five Players and Beyond.","authors":"R. Ford, James Grime, Eric Harshbarger, Brian Pollock","doi":"10.2478/rmm-2023-0004","DOIUrl":"https://doi.org/10.2478/rmm-2023-0004","url":null,"abstract":"Abstract Before a game begins, the players need to decide the order of play. This order of play is determined by each player rolling a die. Does there exist a set of dice such that draws are excluded and each order of play is equally likely? For four players the solution involves four 12-sided dice, sold commercially as Go First Dice. However, the solution for five players remained an open question. We present two solutions. The first solution has a particular mathematical structure known as binary dice, and results in a set of five 60-sided dice, where every place is equally likely. The second solution is an inductive construction that results in one one 36-sided die; two 48-sided dice; one 54-sided die; and one 20-sided die, where each permutation is equally likely.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124316313","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 present several binary clocks. Using different geometric figures, we show how one can devise various novel ways of displaying time. We accompany each design with the mathematical background necessary to understand why these designs work.
{"title":"How to Read a Clock","authors":"G. Teşeleanu, Simion Stoilow","doi":"10.2478/rmm-2023-0005","DOIUrl":"https://doi.org/10.2478/rmm-2023-0005","url":null,"abstract":"Abstract In this paper we present several binary clocks. Using different geometric figures, we show how one can devise various novel ways of displaying time. We accompany each design with the mathematical background necessary to understand why these designs work.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127751557","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 We study a very simple 2-player board game called Dodgem, curiously the game is difficult to analyze when the number of tokens is not the same for the two players. We provide theoretical and experimental elements which indicate which player benefits from the asymmetry of the game.
{"title":"How Unfair is the Unfair Dodgem?","authors":"L. Gerin","doi":"10.2478/rmm-2023-0002","DOIUrl":"https://doi.org/10.2478/rmm-2023-0002","url":null,"abstract":"Abstract We study a very simple 2-player board game called Dodgem, curiously the game is difficult to analyze when the number of tokens is not the same for the two players. We provide theoretical and experimental elements which indicate which player benefits from the asymmetry of the game.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127735927","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 The Wallace-Bolyai-Gerwien theorem states any polygon can be decomposed into a finite number of polygonal pieces that can be translated and rotated to form any polygon of equal area. The theorem was proved in the early 19th century. The minimum number of pieces necessary to form these common dissections remains an open question. In 1905, Henry Dudney demonstrated a four-piece common dissection between a square and equilateral triangle. We investigate the possible existence of a three-piece common dissection. Specifically, we prove that there does not exist a three-piece common dissection using only convex polygons.
{"title":"Economical Dissections","authors":"T. DeGiovanni, Jason Lutz, Katharine Shultis","doi":"10.2478/rmm-2023-0001","DOIUrl":"https://doi.org/10.2478/rmm-2023-0001","url":null,"abstract":"Abstract The Wallace-Bolyai-Gerwien theorem states any polygon can be decomposed into a finite number of polygonal pieces that can be translated and rotated to form any polygon of equal area. The theorem was proved in the early 19th century. The minimum number of pieces necessary to form these common dissections remains an open question. In 1905, Henry Dudney demonstrated a four-piece common dissection between a square and equilateral triangle. We investigate the possible existence of a three-piece common dissection. Specifically, we prove that there does not exist a three-piece common dissection using only convex polygons.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131808990","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 compare the efficiency of the decimal system to the efficiency of different mixed radix representations. We use as a starting point for our study the duodecimal systems suggested by Pascal and the Maya “Long Count” system. Using the quality index we experimentally show that two slight deviations from the duodecimal system are more efficient than the previous two systems and also than the decimal system.
{"title":"Lagrange was Wrong, Pascal was Right","authors":"G. Teşeleanu","doi":"10.2478/rmm-2022-0004","DOIUrl":"https://doi.org/10.2478/rmm-2022-0004","url":null,"abstract":"Abstract In this paper we compare the efficiency of the decimal system to the efficiency of different mixed radix representations. We use as a starting point for our study the duodecimal systems suggested by Pascal and the Maya “Long Count” system. Using the quality index we experimentally show that two slight deviations from the duodecimal system are more efficient than the previous two systems and also than the decimal system.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122545354","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}
Antonella Perucca, Joe Reguengo De Sousa, S. Tronto
Abstract Arithmetic billiards show a nice interplay between arithmetics and geometry. The billiard table is a rectangle with integer side lengths. A pointwise ball moves with constant speed along segments making a 45° angle with the sides and bounces on these. In the classical setting, the ball is shooted from a corner and lands in a corner. We allow the ball to start at any point with integer distances from the sides: either the ball lands in a corner or the trajectory is periodic. The length of the path and of certain segments in the path are precisely (up to the factor √2 or 2√2) the least common multiple and the greatest common divisor of the side lengths.
{"title":"Arithmetic Billiards","authors":"Antonella Perucca, Joe Reguengo De Sousa, S. Tronto","doi":"10.2478/rmm-2022-0003","DOIUrl":"https://doi.org/10.2478/rmm-2022-0003","url":null,"abstract":"Abstract Arithmetic billiards show a nice interplay between arithmetics and geometry. The billiard table is a rectangle with integer side lengths. A pointwise ball moves with constant speed along segments making a 45° angle with the sides and bounces on these. In the classical setting, the ball is shooted from a corner and lands in a corner. We allow the ball to start at any point with integer distances from the sides: either the ball lands in a corner or the trajectory is periodic. The length of the path and of certain segments in the path are precisely (up to the factor √2 or 2√2) the least common multiple and the greatest common divisor of the side lengths.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"31 12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114780312","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 The Spirograph is an old and popular toy that produces aesthetically pleasing and fascinating spiral figures. But are spirals all it can make? In playing with a software implementation of the toy, the author chanced upon a variety of shapes that it can make that are different from its well-known repertoire of spirals, in particular, shapes that have a visible flatness and not the curved spiral geometry that we are accustomed to seeing from the Spirograph. This paper reports on these explorations by the author and his delightful discovery of very elegant and simple geometric relationships between the Spirograph’s structural parameters that enable those patterns.
{"title":"Flattening the Curve. . . of Spirographs","authors":"R. Konkar","doi":"10.2478/rmm-2022-0001","DOIUrl":"https://doi.org/10.2478/rmm-2022-0001","url":null,"abstract":"Abstract The Spirograph is an old and popular toy that produces aesthetically pleasing and fascinating spiral figures. But are spirals all it can make? In playing with a software implementation of the toy, the author chanced upon a variety of shapes that it can make that are different from its well-known repertoire of spirals, in particular, shapes that have a visible flatness and not the curved spiral geometry that we are accustomed to seeing from the Spirograph. This paper reports on these explorations by the author and his delightful discovery of very elegant and simple geometric relationships between the Spirograph’s structural parameters that enable those patterns.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"229 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114602680","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 We define the queens (resp., rooks) diameter-separation number to be the minimum number of pawns for which some placement of those pawns on an n × n board produces a board with a queens graph (resp., rooks graph) with a desired diameter d. We determine these numbers for some small values of d.
{"title":"Diameter-Separation of Chessboard Graphs","authors":"D. Chatham","doi":"10.2478/rmm-2021-0008","DOIUrl":"https://doi.org/10.2478/rmm-2021-0008","url":null,"abstract":"Abstract We define the queens (resp., rooks) diameter-separation number to be the minimum number of pawns for which some placement of those pawns on an n × n board produces a board with a queens graph (resp., rooks graph) with a desired diameter d. We determine these numbers for some small values of d.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123044699","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 This paper presents the total time required to mow a two-dimensional rectangular region of grass using a push mower. In deriving the total time, each of the three ‘well known’ (or intuitive) mowing patterns to cut the entire rectangular grass area is used. Using basic mathematics, analytical and empirical time results for each of the three patterns taken to completely cover this rectangular region are presented, and examples are used to determine which pattern provides an optimal total time to cut a planar rectangular region. This paper provides quantitative information to aid in deciding which mowing pattern to use when cutting one’s lawn.
{"title":"A Simple Guide to Lawn Mowing","authors":"Perry Y.C. Lee, Joshua B. Lee","doi":"10.2478/rmm-2021-0004","DOIUrl":"https://doi.org/10.2478/rmm-2021-0004","url":null,"abstract":"Abstract This paper presents the total time required to mow a two-dimensional rectangular region of grass using a push mower. In deriving the total time, each of the three ‘well known’ (or intuitive) mowing patterns to cut the entire rectangular grass area is used. Using basic mathematics, analytical and empirical time results for each of the three patterns taken to completely cover this rectangular region are presented, and examples are used to determine which pattern provides an optimal total time to cut a planar rectangular region. This paper provides quantitative information to aid in deciding which mowing pattern to use when cutting one’s lawn.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"21 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":"125717266","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 If water is flowing at the same constant rate through each of H ⩾3 hoses, so that any one hose will fill any one of J ⩾ 2 available jars in exactly one hour, then what are the fillable fractions of a jar, and what are the measurable fractions of an hour? Learning to systematically answer such questions will not only equip readers to fluently use fractions, but also introduce or reintroduce them gently to the Queen of Mathematics – Number Theory.
{"title":"Filling Jars to Measure Time","authors":"Tiffany-Chau Le, J. Sarkar","doi":"10.2478/rmm-2021-0001","DOIUrl":"https://doi.org/10.2478/rmm-2021-0001","url":null,"abstract":"Abstract If water is flowing at the same constant rate through each of H ⩾3 hoses, so that any one hose will fill any one of J ⩾ 2 available jars in exactly one hour, then what are the fillable fractions of a jar, and what are the measurable fractions of an hour? Learning to systematically answer such questions will not only equip readers to fluently use fractions, but also introduce or reintroduce them gently to the Queen of Mathematics – Number Theory.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"18 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":"117040504","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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