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Detalles de la revista
Formato
Revista
eISSN
2182-1976
Publicado por primera vez
16 Apr 2016
Periodo de publicación
2 veces al año
Idiomas
Inglés

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Volumen 4 (2017): Edición 8 (December 2017)

Detalles de la revista
Formato
Revista
eISSN
2182-1976
Publicado por primera vez
16 Apr 2016
Periodo de publicación
2 veces al año
Idiomas
Inglés

Buscar

4 Artículos
Acceso abierto

"A difficult case": Pacioli and Cardano on the Chinese Rings

Publicado en línea: 11 Jan 2018
Páginas: 5 - 23

Resumen

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 The first is by Luca Pacioli and was previously unpublished.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.

Palabras clave

  • Chinese rings
  • recreational problems
  • Pacioli
  • Cardano
Acceso abierto

Independence and domination on shogiboard graphs

Publicado en línea: 11 Jan 2018
Páginas: 25 - 37

Resumen

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.

Palabras clave

  • shogi
  • n-queens problem
  • combinatorics
Acceso abierto

A generalization of Trenkler’s magic cubes formula

Publicado en línea: 11 Jan 2018
Páginas: 39 - 45

Resumen

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.

Palabras clave

  • Magic cube
  • regular magic cube
  • magic cube formula
  • Trenkler’s formula
Acceso abierto

Rules for folding polyminoes from one level to two levels

Publicado en línea: 11 Jan 2018
Páginas: 47 - 58

Resumen

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.

Palabras clave

  • Folding polyominoes
4 Artículos
Acceso abierto

"A difficult case": Pacioli and Cardano on the Chinese Rings

Publicado en línea: 11 Jan 2018
Páginas: 5 - 23

Resumen

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.

Palabras clave

  • Chinese rings
  • recreational problems
  • Pacioli
  • Cardano
Acceso abierto

Independence and domination on shogiboard graphs

Publicado en línea: 11 Jan 2018
Páginas: 25 - 37

Resumen

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.

Palabras clave

  • shogi
  • n-queens problem
  • combinatorics
Acceso abierto

A generalization of Trenkler’s magic cubes formula

Publicado en línea: 11 Jan 2018
Páginas: 39 - 45

Resumen

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.

Palabras clave

  • Magic cube
  • regular magic cube
  • magic cube formula
  • Trenkler’s formula
Acceso abierto

Rules for folding polyminoes from one level to two levels

Publicado en línea: 11 Jan 2018
Páginas: 47 - 58

Resumen

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.

Palabras clave

  • Folding polyominoes

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