rss_2.0Recreational Mathematics Magazine FeedSciendo RSS Feed for Recreational Mathematics Magazine Mathematics Magazine Feed to Read a Clock<abstract> <title style='display:none'>Abstract</title> <p>In this paper we present several binary clocks. Using different geometric figures, we show how one can devise various novel ways of displaying time. We accompany each design with the mathematical background necessary to understand why these designs work.</p> </abstract>ARTICLEtrue Dissections<abstract> <title style='display:none'>Abstract</title> <p>The Wallace-Bolyai-Gerwien theorem states any polygon can be decomposed into a finite number of polygonal pieces that can be translated and rotated to form any polygon of equal area. The theorem was proved in the early 19th century. The minimum number of pieces necessary to form these common dissections remains an open question. In 1905, Henry Dudney demonstrated a four-piece common dissection between a square and equilateral triangle. We investigate the possible existence of a three-piece common dissection. Specifically, we prove that there does not exist a three-piece common dissection using only convex polygons.</p> </abstract>ARTICLEtrue First Dice for Five Players and Beyond.<abstract> <title style='display:none'>Abstract</title> <p>Before a game begins, the players need to decide the order of play. This order of play is determined by each player rolling a die. Does there exist a set of dice such that draws are excluded and each order of play is equally likely? For four players the solution involves four 12-sided dice, sold commercially as Go First Dice. However, the solution for five players remained an open question. We present two solutions. The first solution has a particular mathematical structure known as binary dice, and results in a set of five 60-sided dice, where every place is equally likely. The second solution is an inductive construction that results in one one 36-sided die; two 48-sided dice; one 54-sided die; and one 20-sided die, where each permutation is equally likely.</p> </abstract>ARTICLEtrue Unfair is the Unfair Dodgem?<abstract> <title style='display:none'>Abstract</title> <p>We study a very simple 2-player board game called Dodgem, curiously the game is difficult to analyze when the number of tokens is not the same for the two players. We provide theoretical and experimental elements which indicate which player benefits from the asymmetry of the game.</p> </abstract>ARTICLEtrue with Latin Squares<abstract> <title style='display:none'>Abstract</title> <p>Do you want to know what an anti-chiece Latin square is? Or what a non-consecutive toroidal modular Latin square is? We invented a ton of new types of Latin squares, some inspired by existing Sudoku variations. We can’t wait to introduce them to you and answer important questions, such as: do they even exist? If so, under what conditions? What are some of their interesting properties? And how do we generate them?</p> </abstract>ARTICLEtrue Billiards<abstract> <title style='display:none'>Abstract</title> <p>Arithmetic billiards show a nice interplay between arithmetics and geometry. The billiard table is a rectangle with integer side lengths. A pointwise ball moves with constant speed along segments making a 45° angle with the sides and bounces on these. In the classical setting, the ball is shooted from a corner and lands in a corner. We allow the ball to start at any point with integer distances from the sides: either the ball lands in a corner or the trajectory is periodic. The length of the path and of certain segments in the path are precisely (up to the factor √2 or 2√2) the least common multiple and the greatest common divisor of the side lengths.</p> </abstract>ARTICLEtrue Modes, their Associated Chords and their Musicality<abstract> <title style='display:none'>Abstract</title> <p>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.</p> </abstract>ARTICLEtrue the Curve. . . of Spirographs<abstract> <title style='display:none'>Abstract</title> <p>The Spirograph is an old and popular toy that produces aesthetically pleasing and fascinating spiral figures. But are spirals all it can make? In playing with a software implementation of the toy, the author chanced upon a variety of shapes that it can make that are different from its well-known repertoire of spirals, in particular, shapes that have a visible flatness and not the curved spiral geometry that we are accustomed to seeing from the Spirograph. This paper reports on these explorations by the author and his delightful discovery of very elegant and simple geometric relationships between the Spirograph’s structural parameters that enable those patterns.</p> </abstract>ARTICLEtrue was Wrong, Pascal was Right<abstract> <title style='display:none'>Abstract</title> <p>In this paper we compare the efficiency of the decimal system to the efficiency of different mixed radix representations. We use as a starting point for our study the duodecimal systems suggested by Pascal and the Maya “Long Count” system. Using the quality index we experimentally show that two slight deviations from the duodecimal system are more efficient than the previous two systems and also than the decimal system.</p> </abstract>ARTICLEtrue<abstract> <title style='display:none'>Abstract</title> <p>Dots-and-Boxes is a popular children’s game whose winning strategies have been studied by Berlekamp, Conway, Guy, and others. In this article we consider two variations, Dots-and-Triangles and Dots-and-Polygons, both of which utilize the same lattice game board structure as Dots-and-Boxes. The nature of these variations along with this lattice structure lends itself to applying Pick’s theorem to calculate claimed area. Several strategies similar to those studied in Dots-and-Boxes are used to analyze these new variations.</p> </abstract>ARTICLEtrue Jars to Measure Time<abstract> <title style='display:none'>Abstract</title> <p>If water is flowing at the same constant rate through each of <italic>H</italic> ⩾3 hoses, so that any one hose will fill any one of <italic>J</italic> ⩾ 2 available jars in exactly one hour, then what are the fillable fractions of a jar, and what are the measurable fractions of an hour? Learning to systematically answer such questions will not only equip readers to fluently use fractions, but also introduce or reintroduce them gently to the Queen of Mathematics – Number Theory.</p> </abstract>ARTICLEtrue Unequal Chance to a Coin Game Dance: Variants of Penney’s Game<abstract> <title style='display:none'>Abstract</title> <p>We start by exploring and analyzing the various aspects of Penney’s game, examining its possible outcomes as well as its fairness (or lack thereof). In search of a fairer game, we create many variations of the original Penney’s game by altering its rules. Specifically, we introduce the Head-Start Penney’s game, the Post-a-Bobalyptic Penney’s game, the Second-Occurrence Penney’s game, the Two-Coin game, the No-Flippancy game, and the Blended game. We then analyze each of these games and the odds of winning for both players.</p> </abstract>ARTICLEtrue Five-Button Door Lock – Experiment and Discovery in Mathematics Simple Guide to Lawn Mowing<abstract> <title style='display:none'>Abstract</title> <p>This paper presents the total time required to mow a two-dimensional rectangular region of grass using a push mower. In deriving the total time, each of the three ‘well known’ (or intuitive) mowing patterns to cut the entire rectangular grass area is used. Using basic mathematics, analytical and empirical time results for each of the three patterns taken to completely cover this rectangular region are presented, and examples are used to determine which pattern provides an optimal total time to cut a planar rectangular region. This paper provides quantitative information to aid in deciding which mowing pattern to use when cutting one’s lawn.</p> </abstract>ARTICLEtrue Game of Poker Chips, Dominoes and Survival<abstract> <title style='display:none'>Abstract</title> <p>The Game of Poker Chips, Dominoes and Survival fosters team building and high level cooperation in large groups, and is a tool applied in management training exercises. Each player, initially given two colored poker chips, is allowed to make exchanges with the game coordinator according to two rules, and must secure a domino before time is called in order to ‘survive’. Though the rules are simple, it is not evident by their form that the survival of the entire group requires that they cooperate at a high level. From the point of view of the game coordinator, the di culty of the game for the group can be controlled not only by the time limit, but also by the initial distribution of chips, in a way we make precise by a time complexity type argument. That analysis also provides insight into good strategies for group survival, those taking the least amount of time. In addition, coordinators may also want to be aware of when the game is ‘solvable’, that is, when their initial distribution of chips permits the survival of all group members if given su cient time to make exchanges. It turns out that the game is solvable if and only if the initial distribution contains seven chips that have one of two particular color distributions. In addition to being a lively game to play in management training or classroom settings, the analysis of the game after play can make for an engaging exercise in any discrete mathematics course to give a basic introduction to elements of game theory, logical reasoning, number theory and the computation of algorithmic complexities.</p> </abstract>ARTICLEtrue alternative algorithm for the –Queens puzzle<abstract> <title style='display:none'>Abstract</title> <p>In this paper a new method for solving the problem of placing <italic>n</italic> queens on a <italic>n×n</italic> chessboard such that no two queens directly threaten one another and considering that several immovable queens are already occupying established positions on the board is presented. At first, it is applied to the 8–Queens puzzle on a classical chessboard and finally to the <italic>n</italic> Queens completion puzzle. Furthermore, this method allows finding repetitive patterns of solutions for any <italic>n</italic>.</p> </abstract>ARTICLEtrue Coins<abstract> <title style='display:none'>Abstract</title> <p>In this paper we consider a scenario where there are several algorithms for solving a given problem. Each algorithm is associated with a probability of success and a cost, and there is also a penalty for failing to solve the problem. The user may run one algorithm at a time for the specified cost, or give up and pay the penalty. The probability of success may be implied by randomization in the algorithm, or by assuming a probability distribution on the input space, which lead to different variants of the problem. The goal is to minimize the expected cost of the process under the assumption that the algorithms are independent. We study several variants of this problem, and present possible solution strategies and a hardness result.</p> </abstract>ARTICLEtrue of Chessboard Graphs<abstract> <title style='display:none'>Abstract</title> <p>We define the <italic>queens (resp., rooks) diameter-separation number</italic> to be the minimum number of pawns for which some placement of those pawns on an <italic>n</italic> × <italic>n</italic> board produces a board with a queens graph (resp., rooks graph) with a desired diameter <italic>d</italic>. We determine these numbers for some small values of <italic>d</italic>.</p> </abstract>ARTICLEtrue Independence on the Surface of a Square Prism<abstract> <title style='display:none'>Abstract</title> <p>Bishop Independence concerns determining the maximum number of bishops that can be placed on a board such that no bishop can attack any other bishop. This paper presents the solution to the bishop independence problem, determining the bishop independence number, for all sizes of boards on the surface of a square prism.</p> </abstract>ARTICLEtrue is Guilty?<abstract> <title style='display:none'>Abstract</title> <p>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?”</p> </abstract>ARTICLEtrue