Abstract In this study, we define vedic cube as the layout of each digital root in a three-dimensional multiplication table. In order to discover the geometric patterns in vedic cube, we adopt two methods to analyze the digital root in a three-dimensional space. The first method is floor method, which divides vedic cube into several X-Y planes according to different Z values (floors) to analyze the geometric characteristics on each floor. The second method is symmetric plane method, which decomposes vedic cube by its main and secondary symmetric planes.
{"title":"Digital Root Patterns of Three-Dimensional Space","authors":"Chia-Yu Lin","doi":"10.1515/rmm-2016-0002","DOIUrl":"https://doi.org/10.1515/rmm-2016-0002","url":null,"abstract":"Abstract In this study, we define vedic cube as the layout of each digital root in a three-dimensional multiplication table. In order to discover the geometric patterns in vedic cube, we adopt two methods to analyze the digital root in a three-dimensional space. The first method is floor method, which divides vedic cube into several X-Y planes according to different Z values (floors) to analyze the geometric characteristics on each floor. The second method is symmetric plane method, which decomposes vedic cube by its main and secondary symmetric planes.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129462916","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}
Alda Carvalho, Carlos Eduardo Pereira dos Santos, J. Silva
Abstract In this work, we present a mathematical interpretation for the masterpiece Allégorie de la Géométrie (1649), painted by the French baroque artist Laurent de La Hyre (1606-1656)
{"title":"Allégorie de la Géométrie. A Mathematical Interpretation","authors":"Alda Carvalho, Carlos Eduardo Pereira dos Santos, J. Silva","doi":"10.1515/rmm-2016-0003","DOIUrl":"https://doi.org/10.1515/rmm-2016-0003","url":null,"abstract":"Abstract In this work, we present a mathematical interpretation for the masterpiece Allégorie de la Géométrie (1649), painted by the French baroque artist Laurent de La Hyre (1606-1656)","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124593283","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 new idea for a binary clock is presented. It displays the time using a triangular array of 15 lamps each representing a certain amount of time. It is shown that such a geometric, triangular arrangement is only possible because our system of time divisions is based on a sexagesimal system in which the num- ber of minutes in 12 hours equals the factorial of a natural number (720 = 6!). An interactive applet allows one to “play” with the clock.
{"title":"The Triangular Binary Clock","authors":"J. Pretz","doi":"10.1515/rmm-2016-0001","DOIUrl":"https://doi.org/10.1515/rmm-2016-0001","url":null,"abstract":"Abstract A new idea for a binary clock is presented. It displays the time using a triangular array of 15 lamps each representing a certain amount of time. It is shown that such a geometric, triangular arrangement is only possible because our system of time divisions is based on a sexagesimal system in which the num- ber of minutes in 12 hours equals the factorial of a natural number (720 = 6!). An interactive applet allows one to “play” with the clock.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131453996","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 simple and popular childhood game, loves or the love calculator, involves an iterated rule applied to a string of digits and gives rise to surprisingly rich behaviour. Traditionally, players’ names are used to set the initial conditions for an instance of the game: its behaviour for an exhaustive set of pairings of popular UK childrens’ names, and for more general initial conditions, is examined. Convergence to a fixed outcome (the desired result) is not guaranteed, even for some plausible first name pairings. No pairs of top-50 common first names exhibit non-convergence, suggesting that it is rare in the playground; however, including surnames makes non-convergence more likely due to higher letter counts (for example, “Reese Witherspoon loves Calvin Harris”). Difierent game keywords (including from difierent languages) are also considered. An estimate for non-convergence propensity is derived: if the sum m of digits in a string of length w obeys m > 18=(3=2/)w-4, convergence is less likely. Pairs of top UK names with pairs of ‘O’s and several ‘L’s (for example, Chloe and Joseph, or Brooke and Scarlett) often attain high scores. When considering individual names playing with a range of partners, those with no loves letters score lowest, and names with intermediate (not simply the highest) letter counts often perform best, with Connor and Evie averaging the highest scores when played with other UK top names.
{"title":"Endless Love: On the Termination of a Playground Number Game","authors":"Iain G. Johnston","doi":"10.1515/rmm-2016-0005","DOIUrl":"https://doi.org/10.1515/rmm-2016-0005","url":null,"abstract":"Abstract A simple and popular childhood game, loves or the love calculator, involves an iterated rule applied to a string of digits and gives rise to surprisingly rich behaviour. Traditionally, players’ names are used to set the initial conditions for an instance of the game: its behaviour for an exhaustive set of pairings of popular UK childrens’ names, and for more general initial conditions, is examined. Convergence to a fixed outcome (the desired result) is not guaranteed, even for some plausible first name pairings. No pairs of top-50 common first names exhibit non-convergence, suggesting that it is rare in the playground; however, including surnames makes non-convergence more likely due to higher letter counts (for example, “Reese Witherspoon loves Calvin Harris”). Difierent game keywords (including from difierent languages) are also considered. An estimate for non-convergence propensity is derived: if the sum m of digits in a string of length w obeys m > 18=(3=2/)w-4, convergence is less likely. Pairs of top UK names with pairs of ‘O’s and several ‘L’s (for example, Chloe and Joseph, or Brooke and Scarlett) often attain high scores. When considering individual names playing with a range of partners, those with no loves letters score lowest, and names with intermediate (not simply the highest) letter counts often perform best, with Connor and Evie averaging the highest scores when played with other UK top names.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128477833","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 introduce a two player game on an n × n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the parity of the number of queens already on the board which can attack that square. The game is explored as well as its variations and complexity.
{"title":"Exploring mod 2 n-queens games","authors":"Tricia Muldoon Brown, Abrahim Ladha","doi":"10.2478/rmm-2019-0002","DOIUrl":"https://doi.org/10.2478/rmm-2019-0002","url":null,"abstract":"Abstract We introduce a two player game on an n × n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the parity of the number of queens already on the board which can attack that square. The game is explored as well as its variations and complexity.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115704017","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}
{"title":"Demystifying Benjamin Franklin’s Other 8-Square","authors":"M. Ahmed","doi":"10.1515/rmm-2017-0012","DOIUrl":"https://doi.org/10.1515/rmm-2017-0012","url":null,"abstract":"Abstract In this article, we reveal how Benjamin Franklin constructed his second 8 × 8 magic square. We also construct two new 8 × 8 Franklin squares","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116114223","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 By using two different invariants for the Rubik’s Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing the target shapes to specific configurations: the octominoid 3D shapes, with all tiles parallel to one coordinate plane; and the planar “face-up” shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles. There are still plenty of interesting configurations that do not belong to either of these two collections. The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants. We were able to actually construct all octominoid shapes with vanishing invariants and most of the planar “face-up” configurations. Particularly important is the topological invariant, of which we recently found mention in [7] by Tom Verhoeff.
{"title":"Exploring the “Rubik's Magic” Universe","authors":"M. Paolini","doi":"10.1515/rmm-2017-0013","DOIUrl":"https://doi.org/10.1515/rmm-2017-0013","url":null,"abstract":"Abstract By using two different invariants for the Rubik’s Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing the target shapes to specific configurations: the octominoid 3D shapes, with all tiles parallel to one coordinate plane; and the planar “face-up” shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles. There are still plenty of interesting configurations that do not belong to either of these two collections. The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants. We were able to actually construct all octominoid shapes with vanishing invariants and most of the planar “face-up” configurations. Particularly important is the topological invariant, of which we recently found mention in [7] by Tom Verhoeff.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115261302","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 a mathematical way of defining musical modes and we define the musicality of a mode as a product of three diferent factors. We conclude by classyfing the modes which are most musical according to our definition.
{"title":"Musical Modes, their Associated Chords and their Musicality","authors":"M. Cocos, Kent Kidman","doi":"10.2478/rmm-2022-0005","DOIUrl":"https://doi.org/10.2478/rmm-2022-0005","url":null,"abstract":"Abstract In this paper we present a mathematical way of defining musical modes and we define the musicality of a mode as a product of three diferent factors. We conclude by classyfing the modes which are most musical according to our definition.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"10 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132032070","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 Tangram is a puzzle in which seven tiles are arranged to make various shapes. Four families of tangram shapes have been mathematically defined and their members enumerated. This paper defines a fifth family, enumerates its members, explains its taxonomic relationship with the previously-defined families, and provides some interesting examples
{"title":"Star Tangrams","authors":"R. Graber, S. Pollard, R. Read","doi":"10.1515/rmm-2016-0004","DOIUrl":"https://doi.org/10.1515/rmm-2016-0004","url":null,"abstract":"Abstract The Tangram is a puzzle in which seven tiles are arranged to make various shapes. Four families of tangram shapes have been mathematically defined and their members enumerated. This paper defines a fifth family, enumerates its members, explains its taxonomic relationship with the previously-defined families, and provides some interesting examples","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121636653","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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