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Détails du magazine
Format
Magazine
eISSN
2182-1976
Première publication
16 Apr 2016
Période de publication
2 fois par an
Langues
Anglais

Chercher

Volume 4 (2017): Edition 7 (May 2017)

Détails du magazine
Format
Magazine
eISSN
2182-1976
Première publication
16 Apr 2016
Période de publication
2 fois par an
Langues
Anglais

Chercher

6 Articles

Games and Puzzles

Accès libre

Several Bounds for the K-Tower of Hanoi Puzzle

Publié en ligne: 09 Jun 2017
Pages: 79 - 118

Résumé

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.

Mots clés

  • Tower of Hanoi Puzzle
  • minimum number of moves
Accès libre

Eternal Picaria

Publié en ligne: 09 Jun 2017
Pages: 119 - 133

Résumé

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.

Mots clés

  • Abstract strategy game
  • Alignment game
  • Board game
  • Cyclic game
  • Loopy game
  • Luffarschack
  • Play-proof
  • Tapatan
  • Three-in-a-row
  • Three men’s morris
  • Zuni tribe
Accès libre

Exploring the “Rubik's Magic” Universe

Publié en ligne: 09 Jun 2017
Pages: 29 - 64

Résumé

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.

Mots clés

  • Rubik's Magic puzzle
  • octominoid 3D shapes
  • topological invariants
Accès libre

Designing Peg Solitaire Puzzles

Publié en ligne: 09 Jun 2017
Pages: 5 - 19

Résumé

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.

Mots clés

  • Solitaire puzzles
  • game design

Articles

Accès libre

Demystifying Benjamin Franklin’s Other 8-Square

Publié en ligne: 09 Jun 2017
Pages: 21 - 27

Résumé

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

Mots clés

  • Magic squares
  • Benjamin Franklin’s 8-squares

Mathmagic

Accès libre

On a Mathematical Model for an Old Card Trick

Publié en ligne: 09 Jun 2017
Pages: 65 - 77

Résumé

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.

Mots clés

  • Modular arithmetic
  • fixed point theory of discrete functions
  • three-pile trick
6 Articles

Games and Puzzles

Accès libre

Several Bounds for the K-Tower of Hanoi Puzzle

Publié en ligne: 09 Jun 2017
Pages: 79 - 118

Résumé

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.

Mots clés

  • Tower of Hanoi Puzzle
  • minimum number of moves
Accès libre

Eternal Picaria

Publié en ligne: 09 Jun 2017
Pages: 119 - 133

Résumé

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.

Mots clés

  • Abstract strategy game
  • Alignment game
  • Board game
  • Cyclic game
  • Loopy game
  • Luffarschack
  • Play-proof
  • Tapatan
  • Three-in-a-row
  • Three men’s morris
  • Zuni tribe
Accès libre

Exploring the “Rubik's Magic” Universe

Publié en ligne: 09 Jun 2017
Pages: 29 - 64

Résumé

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.

Mots clés

  • Rubik's Magic puzzle
  • octominoid 3D shapes
  • topological invariants
Accès libre

Designing Peg Solitaire Puzzles

Publié en ligne: 09 Jun 2017
Pages: 5 - 19

Résumé

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.

Mots clés

  • Solitaire puzzles
  • game design

Articles

Accès libre

Demystifying Benjamin Franklin’s Other 8-Square

Publié en ligne: 09 Jun 2017
Pages: 21 - 27

Résumé

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

Mots clés

  • Magic squares
  • Benjamin Franklin’s 8-squares

Mathmagic

Accès libre

On a Mathematical Model for an Old Card Trick

Publié en ligne: 09 Jun 2017
Pages: 65 - 77

Résumé

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.

Mots clés

  • Modular arithmetic
  • fixed point theory of discrete functions
  • three-pile trick

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