Abstract The three-pile trick is a well-known card trick performed with a deck of 27 cards which dates back to the early seventeenth century at least and its objective is to uncover the card chosen by a volunteer. The main purpose of this research is to give a mathematical generalization of the three-pile trick for any deck of ab cards with a, b ≥ 2 any integers by means of a finite family of simple discrete functions. Then, it is proved each of these functions has just one or two stable fixed points. Based on this findings a list of 222 (three-pile trick)-type brand new card tricks was generated for either a package of 52 playing cards or any appropriate portion of it with a number of piles between 3 and 7. It is worth noting that all the card tricks on the list share the three main properties that have characterized the three-pile trick: simplicity, self-performing and infallibility. Finally, a general performing protocol, useful for magicians, is given for all the cases. All the employed math techniques involve naive theory of discrete functions, basic properties of the quotient and remainder of the division of integers and modular arithmetic.
{"title":"On a Mathematical Model for an Old Card Trick","authors":"Roy Quintero","doi":"10.1515/rmm-2017-0014","DOIUrl":"https://doi.org/10.1515/rmm-2017-0014","url":null,"abstract":"Abstract The three-pile trick is a well-known card trick performed with a deck of 27 cards which dates back to the early seventeenth century at least and its objective is to uncover the card chosen by a volunteer. The main purpose of this research is to give a mathematical generalization of the three-pile trick for any deck of ab cards with a, b ≥ 2 any integers by means of a finite family of simple discrete functions. Then, it is proved each of these functions has just one or two stable fixed points. Based on this findings a list of 222 (three-pile trick)-type brand new card tricks was generated for either a package of 52 playing cards or any appropriate portion of it with a number of piles between 3 and 7. It is worth noting that all the card tricks on the list share the three main properties that have characterized the three-pile trick: simplicity, self-performing and infallibility. Finally, a general performing protocol, useful for magicians, is given for all the cases. All the employed math techniques involve naive theory of discrete functions, basic properties of the quotient and remainder of the division of integers and modular arithmetic.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"50 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":"115945221","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 Polyominoes have been the focus of many recreational and research investigations. In this article, the authors investigate whether a paper cutout of a polyomino can be folded to produce a second polyomino in the same shape as the original, but now with two layers of paper. For the folding, only “corner folds” and “half edge cuts” are allowed, unless the polyomino forms a closed loop, in which case one is allowed to completely cut two squares in the polyomino apart. With this set of allowable moves, the authors present algorithms for folding different types of polyominoes and prove that certain polyominoes can successfully be folded to two layers. The authors also establish that other polyominoes cannot be folded to two layers if only these moves are allowed.
{"title":"Rules for folding polyminoes from one level to two levels","authors":"Julia Martín, Elizabeth Wilcox","doi":"10.1515/rmm-2017-0020","DOIUrl":"https://doi.org/10.1515/rmm-2017-0020","url":null,"abstract":"Abstract Polyominoes have been the focus of many recreational and research investigations. In this article, the authors investigate whether a paper cutout of a polyomino can be folded to produce a second polyomino in the same shape as the original, but now with two layers of paper. For the folding, only “corner folds” and “half edge cuts” are allowed, unless the polyomino forms a closed loop, in which case one is allowed to completely cut two squares in the polyomino apart. With this set of allowable moves, the authors present algorithms for folding different types of polyominoes and prove that certain polyominoes can successfully be folded to two layers. The authors also establish that other polyominoes cannot be folded to two layers if only these moves are allowed.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132576815","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 Sudoku grids are often cited as being useful in cryptography as a key for some encryption process. Historically transporting keys over an alternate channel has been very difficult. This article describes how a Sudoku grid key can be secretly transported using quantum key distribution methods whereby partial grid (or puzzle) can be received and the full key can be recreated by solving the puzzle.
{"title":"Quantum Distribution of a Sudoku Key","authors":"Sian K. Jones","doi":"10.1515/rmm-2016-0009","DOIUrl":"https://doi.org/10.1515/rmm-2016-0009","url":null,"abstract":"Abstract Sudoku grids are often cited as being useful in cryptography as a key for some encryption process. Historically transporting keys over an alternate channel has been very difficult. This article describes how a Sudoku grid key can be secretly transported using quantum key distribution methods whereby partial grid (or puzzle) can be received and the full key can be recreated by solving the puzzle.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120954750","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 discuss in detail what is behind April Fool’s Day
在本文中，我们详细讨论了愚人节的背后是什么
{"title":"Where are (Pseudo)Science Fool’s Hoax Articles in April From?","authors":"Tereza Bártlová","doi":"10.1515/rmm-2016-0007","DOIUrl":"https://doi.org/10.1515/rmm-2016-0007","url":null,"abstract":"Abstract In this paper, we discuss in detail what is behind April Fool’s Day","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124102991","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 classic n-queens problem asks for placements of just n mutually non-attacking queens on an n × n board. By adding enough pawns, we can arrange to fill roughly one-quarter of the board with mutually non-attacking queens. How many pawns do we need? We discuss that question for square boards as well as rectangular m × n boards.
{"title":"The Maximum Queens Problem with Pawns","authors":"D. Chatham","doi":"10.1515/rmm-2016-0010","DOIUrl":"https://doi.org/10.1515/rmm-2016-0010","url":null,"abstract":"Abstract The classic n-queens problem asks for placements of just n mutually non-attacking queens on an n × n board. By adding enough pawns, we can arrange to fill roughly one-quarter of the board with mutually non-attacking queens. How many pawns do we need? We discuss that question for square boards as well as rectangular m × n boards.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117090783","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 Peg solitaire is an old puzzle with a 300 year history. We consider two ways a computer can be utilized to find interesting peg solitaire puzzles. It is common for a peg solitaire puzzle to begin from a symmetric board position, we have computed solvable symmetric board positions for four board shapes. A new idea is to search for board positions which have a unique starting jump leading to a solution. We show many challenging puzzles uncovered by this search technique. Clever solvers can take advantage of the uniqueness property to help solve these puzzles.
{"title":"Designing Peg Solitaire Puzzles","authors":"George I. Bell","doi":"10.1515/rmm-2017-0011","DOIUrl":"https://doi.org/10.1515/rmm-2017-0011","url":null,"abstract":"Abstract Peg solitaire is an old puzzle with a 300 year history. We consider two ways a computer can be utilized to find interesting peg solitaire puzzles. It is common for a peg solitaire puzzle to begin from a symmetric board position, we have computed solvable symmetric board positions for four board shapes. A new idea is to search for board positions which have a unique starting jump leading to a solution. We show many challenging puzzles uncovered by this search technique. Clever solvers can take advantage of the uniqueness property to help solve these puzzles.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134243874","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 Picaria is a traditional board game, played by the Zuni tribe of the American Southwest and other parts of the world, such as a rural Southwest region in Sweden. It is related to the popular children’s game of Tic-tac-toe, but the 2 players have only 3 stones each, and in the second phase of the game, pieces are slided, along specified move edges, in attempts to create the three-in-a-row. We provide a rigorous solution, and prove that the game is a draw; moreover our solution gives insights to strategies that players can use.
{"title":"Eternal Picaria","authors":"U. Larsson, Israel Rocha","doi":"10.1515/rmm-2017-0016","DOIUrl":"https://doi.org/10.1515/rmm-2017-0016","url":null,"abstract":"Abstract Picaria is a traditional board game, played by the Zuni tribe of the American Southwest and other parts of the world, such as a rural Southwest region in Sweden. It is related to the popular children’s game of Tic-tac-toe, but the 2 players have only 3 stones each, and in the second phase of the game, pieces are slided, along specified move edges, in attempts to create the three-in-a-row. We provide a rigorous solution, and prove that the game is a draw; moreover our solution gives insights to strategies that players can use.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132778838","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 dominant part in the mental calculation of the day of the week for any given date is to determine the year share, that is, the contribution of the two-digit year part of the date. This paper describes a number of year share computation methods, some well-known and some new. The “Parity Minus 3” method, in particular, is a new alternative to the popular “Odd+11” method. The paper categorizes the methods of year share computation, and presents simpler proofs of their correctness than usually provided.
{"title":"Finding the Year’s Share in Day-of-Week Calculations","authors":"S. Abdali","doi":"10.1515/rmm-2016-0008","DOIUrl":"https://doi.org/10.1515/rmm-2016-0008","url":null,"abstract":"Abstract The dominant part in the mental calculation of the day of the week for any given date is to determine the year share, that is, the contribution of the two-digit year part of the date. This paper describes a number of year share computation methods, some well-known and some new. The “Parity Minus 3” method, in particular, is a new alternative to the popular “Odd+11” method. The paper categorizes the methods of year share computation, and presents simpler proofs of their correctness than usually provided.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121635957","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 contrast to traditional toy tracks, a patented system allows the creation of a large number of tracks with a minimal number of pieces, and whose loops always close properly. These circuits strongly resemble traditional self-avoiding polygons (whose explicit enumeration has not yet been resolved for an arbitrary number of squares) yet there are numerous differences, notably the fact that the geometric constraints are different than those of self-avoiding polygons. We present the methodology allowing the construction and enumeration of all of the possible tracks containing a given number of pieces. For small numbers of pieces, the exact enumeration will be treated. For greater numbers of pieces, only an estimation will be offered. In the latter case, a randomly construction of circuits is also given. We will give some routes for generalizations for similar problems.
{"title":"Construction and Enumeration of Circuits Capable of Guiding a Miniature Vehicle","authors":"J. Bastien","doi":"10.1515/rmm-2016-0006","DOIUrl":"https://doi.org/10.1515/rmm-2016-0006","url":null,"abstract":"Abstract In contrast to traditional toy tracks, a patented system allows the creation of a large number of tracks with a minimal number of pieces, and whose loops always close properly. These circuits strongly resemble traditional self-avoiding polygons (whose explicit enumeration has not yet been resolved for an arbitrary number of squares) yet there are numerous differences, notably the fact that the geometric constraints are different than those of self-avoiding polygons. We present the methodology allowing the construction and enumeration of all of the possible tracks containing a given number of pieces. For small numbers of pieces, the exact enumeration will be treated. For greater numbers of pieces, only an estimation will be offered. In the latter case, a randomly construction of circuits is also given. We will give some routes for generalizations for similar problems.","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132623052","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}
B. Chen, Ezra Erives, Leon Fan, Michael Gerovitch, Jonathan Hsu, T. Khovanova, Neil Malur, Ashwin Padaki, Nastia Polina, Will Sun, Jacob Tan, Andrew The
Abstract We discuss a generalization of logic puzzles in which truth-tellers and liars are allowed to deviate from their pattern in case of one particular question: “Are you guilty?”
{"title":"Who is Guilty?","authors":"B. Chen, Ezra Erives, Leon Fan, Michael Gerovitch, Jonathan Hsu, T. Khovanova, Neil Malur, Ashwin Padaki, Nastia Polina, Will Sun, Jacob Tan, Andrew The","doi":"10.2478/rmm-2021-0010","DOIUrl":"https://doi.org/10.2478/rmm-2021-0010","url":null,"abstract":"Abstract We discuss a generalization of logic puzzles in which truth-tellers and liars are allowed to deviate from their pattern in case of one particular question: “Are you guilty?”","PeriodicalId":120489,"journal":{"name":"Recreational Mathematics Magazine","volume":"36 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113973478","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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