Abstract In this paper, we define the term Mathematical Sculpture, a task somehow complex. Also, we present a classification of mathematical sculptures as exhaustive and complete as possible. Our idea consists in establishing general groups for different branches of Mathematics, subdividing these groups according to the main mathematical concepts used in the sculpture design.
{"title":"A Classification of Mathematical Sculpture","authors":"Ricardo Zalaya, J. Barrallo","doi":"10.2478/rmm-2018-0004","DOIUrl":"https://doi.org/10.2478/rmm-2018-0004","url":null,"abstract":"Abstract In this paper, we define the term Mathematical Sculpture, a task somehow complex. Also, we present a classification of mathematical sculptures as exhaustive and complete as possible. Our idea consists in establishing general groups for different branches of Mathematics, subdividing these groups according to the main mathematical concepts used in the sculpture design.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114699861","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 Gardner asked whether it was possible to tile/pack the squares 1×1,…, 24×24 in a 70×70 square. Arguments that it is impossible have been given by Bitner–Reingold and more recently by Korf–Mofitt–Pollack. Here we outline a simpler algorithm, which we hope could be used to give an alternative and more direct proof in the future. We also derive results of independent interest concerning such packings.
{"title":"Optimal Rectangle Packing for the 70 Square","authors":"Brian Laverty, T. Murphy","doi":"10.2478/rmm-2018-0001","DOIUrl":"https://doi.org/10.2478/rmm-2018-0001","url":null,"abstract":"Abstract Gardner asked whether it was possible to tile/pack the squares 1×1,…, 24×24 in a 70×70 square. Arguments that it is impossible have been given by Bitner–Reingold and more recently by Korf–Mofitt–Pollack. Here we outline a simpler algorithm, which we hope could be used to give an alternative and more direct proof in the future. We also derive results of independent interest concerning such packings.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129828291","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 Using six colors, one per side, cubes can be colored in 30 unique ways. In this paper, a row and column pattern in Conway’s matrix always leads to a selection of eight cubes to replicate one of the 30 cubes. Each cube in the set of 30 has a 2 × 2 × 2 replica with inside faces of matching color. The eight cubes of each replica can be configured in two different ways.
{"title":"2 × 2 × 2 Color Cubes","authors":"Raymond Siegrist","doi":"10.2478/rmm-2018-0003","DOIUrl":"https://doi.org/10.2478/rmm-2018-0003","url":null,"abstract":"Abstract Using six colors, one per side, cubes can be colored in 30 unique ways. In this paper, a row and column pattern in Conway’s matrix always leads to a selection of eight cubes to replicate one of the 30 cubes. Each cube in the set of 30 has a 2 × 2 × 2 replica with inside faces of matching color. The eight cubes of each replica can be configured in two different ways.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132929089","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 investigate a type of a Sudoku variant called Sudo-Kurve, which allows bent rows and columns, and develop a new, yet equivalent, variant we call a Sudo-Cube. We examine the total number of distinct solution grids for this type with or without symmetry. We study other mathematical aspects of this puzzle along with the minimum number of clues needed and the number of ways to place individual symbols.
{"title":"Mathematics of a Sudo-Kurve","authors":"T. Khovanova, Wayne Zhao","doi":"10.2478/rmm-2018-0005","DOIUrl":"https://doi.org/10.2478/rmm-2018-0005","url":null,"abstract":"Abstract We investigate a type of a Sudoku variant called Sudo-Kurve, which allows bent rows and columns, and develop a new, yet equivalent, variant we call a Sudo-Cube. We examine the total number of distinct solution grids for this type with or without symmetry. We study other mathematical aspects of this puzzle along with the minimum number of clues needed and the number of ways to place individual symbols.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116873491","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 article presents a simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is an ellipse, then the locus of vertices of the cones is a hyperbola. The hyperbola has foci which coincidence with the ellipse vertices. Similarly, if the given curve is the hyperbola, the locus of vertex of the cones is the ellipse. In the second case, the foci of the ellipse are located in the hyperbola’s vertices. These two relationships create a kind of conjunction between the ellipse and the hyperbola which originate from the cones used for generation of these curves. The presented conjunction of the ellipse and hyperbola is a perfect example of mathematical beauty which may be shown by the use of very simple geometry. As in the past the conic curves appear to be very interesting and fruitful mathematical beings.
{"title":"Ellipse, hyperbola and their conjunction","authors":"A. Kobiera","doi":"10.2478/rmm-2018-0006","DOIUrl":"https://doi.org/10.2478/rmm-2018-0006","url":null,"abstract":"Abstract This article presents a simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is an ellipse, then the locus of vertices of the cones is a hyperbola. The hyperbola has foci which coincidence with the ellipse vertices. Similarly, if the given curve is the hyperbola, the locus of vertex of the cones is the ellipse. In the second case, the foci of the ellipse are located in the hyperbola’s vertices. These two relationships create a kind of conjunction between the ellipse and the hyperbola which originate from the cones used for generation of these curves. The presented conjunction of the ellipse and hyperbola is a perfect example of mathematical beauty which may be shown by the use of very simple geometry. As in the past the conic curves appear to be very interesting and fruitful mathematical beings.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122170443","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 Chinese rings puzzle is one of those recreational mathematical problems known for several centuries in the West as well as in Asia. Its origin is diffcult to ascertain but is most likely not Chinese. In this paper we provide an English translation, based on a mathematical analysis of the puzzle, of two sixteenth-century witness accounts. The first is by Luca Pacioli and was previously unpublished. The second is by Girolamo Cardano for which we provide an interpretation considerably different from existing translations. Finally, both treatments of the puzzle are compared, pointing out the presence of an implicit idea of non-numerical recursive algorithms.
{"title":"\"A difficult case\": Pacioli and Cardano on the Chinese Rings","authors":"Albrecht Heeffer, A. M. Hinz","doi":"10.1515/rmm-2017-0017","DOIUrl":"https://doi.org/10.1515/rmm-2017-0017","url":null,"abstract":"Abstract The Chinese rings puzzle is one of those recreational mathematical problems known for several centuries in the West as well as in Asia. Its origin is diffcult to ascertain but is most likely not Chinese. In this paper we provide an English translation, based on a mathematical analysis of the puzzle, of two sixteenth-century witness accounts. The first is by Luca Pacioli and was previously unpublished. The second is by Girolamo Cardano for which we provide an interpretation considerably different from existing translations. Finally, both treatments of the puzzle are compared, pointing out the presence of an implicit idea of non-numerical recursive algorithms.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130759041","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 Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.
{"title":"Independence and domination on shogiboard graphs","authors":"D. Chatham","doi":"10.1515/rmm-2017-0018","DOIUrl":"https://doi.org/10.1515/rmm-2017-0018","url":null,"abstract":"Abstract Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"194 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120888178","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 A Magic Cube of order p is a p×p×p cubical array with non-repeated entries from the set {1, 2, . . . , p3} such that all rows, columns, pillars and space diagonals have the same sum. In this paper, we show that a formula introduced in The Mathematical Gazette 84(2000), by M. Trenkler, for generating odd order magic cubes is a special case of a more general class of formulas. We derive sufficient conditions for the formulas in the new class to generate magic cubes, and we refer to the resulting class as regular magic cubes. We illustrate these ideas by deriving three new formulas that generate magic cubes of odd order that differ from each other and from the magic cubes generated with Trenkler’s rule.
{"title":"A generalization of Trenkler’s magic cubes formula","authors":"L. Uko, T. L. Barron","doi":"10.1515/rmm-2017-0019","DOIUrl":"https://doi.org/10.1515/rmm-2017-0019","url":null,"abstract":"Abstract A Magic Cube of order p is a p×p×p cubical array with non-repeated entries from the set {1, 2, . . . , p3} such that all rows, columns, pillars and space diagonals have the same sum. In this paper, we show that a formula introduced in The Mathematical Gazette 84(2000), by M. Trenkler, for generating odd order magic cubes is a special case of a more general class of formulas. We derive sufficient conditions for the formulas in the new class to generate magic cubes, and we refer to the resulting class as regular magic cubes. We illustrate these ideas by deriving three new formulas that generate magic cubes of odd order that differ from each other and from the magic cubes generated with Trenkler’s rule.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127499745","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}
R. Akiyama, Nozomi Abe, H. Fujita, Y. Inaba, Mari Hataoka, S. Ito, Satomi Seita
Abstract We treat the boundary of the union of blocks in the Jenga game as a surface with a polyhedral structure and consider its genus. We generalize the game and determine the maximum genus among the configurations in the generalized game.
{"title":"Maximum Genus of the Jenga Like Configurations","authors":"R. Akiyama, Nozomi Abe, H. Fujita, Y. Inaba, Mari Hataoka, S. Ito, Satomi Seita","doi":"10.2478/rmm-2018-0002","DOIUrl":"https://doi.org/10.2478/rmm-2018-0002","url":null,"abstract":"Abstract We treat the boundary of the union of blocks in the Jenga game as a surface with a polyhedral structure and consider its genus. We generalize the game and determine the maximum genus among the configurations in the generalized game.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122283142","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}
Stephen B. Gregg, B. Hopkins, Kristi Karber, Thomas Milligan, Johnny Sharp
Abstract We consider special cases of a modified version of the Tower of Hanoi puzzle and demonstrate how to find upper bounds on the minimum number of moves that it takes to complete these cases.
我们考虑了一个修改版本的河内塔谜题的特殊情况，并演示了如何找到完成这些情况所需的最小移动数的上界。
{"title":"Several Bounds for the K-Tower of Hanoi Puzzle","authors":"Stephen B. Gregg, B. Hopkins, Kristi Karber, Thomas Milligan, Johnny Sharp","doi":"10.1515/rmm-2017-0015","DOIUrl":"https://doi.org/10.1515/rmm-2017-0015","url":null,"abstract":"Abstract We consider special cases of a modified version of the Tower of Hanoi puzzle and demonstrate how to find upper bounds on the minimum number of moves that it takes to complete these cases.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122062149","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：