rss_2.0Recreational Mathematics Magazine FeedSciendo RSS Feed for Recreational Mathematics Magazine Mathematics Magazine Feed of mutually non-attacking chess pieces of mixed type<abstract> <title style='display:none'>Abstract</title> <p>We present placements of mutually non-attacking chess pieces of mixed type that occupy more than half of the squares of an <italic>m</italic> × <italic>n</italic> board. If both white and black pawns are allowed as separate types, there are arrangements, which we also present, that occupy at least two-thirds of the board squares.</p> </abstract>ARTICLEtrue numbers and Rithmomachia<abstract> <title style='display:none'>Abstract</title> <p>This paper shows our discovery of the fiboquadratic numbers, quadratic Fibonacci numbers in rithmomachia. This board game consists in performing mathematical operations to attack an opponent through geometric numbered pieces. We will show that the numbers on the board pieces are the first terms of six sequences that we will define through a recursive pattern of superpartients. We then prove that this extension is the solution of a recurrence relation with initial conditions, and therefore, is unique. The fiboquadratic numbers we introduce in this work are not limited to the context of this board. In general, we can generate any fiboquadratic sequence with arbitrary initial conditions.</p> </abstract>ARTICLEtrue Playing With a Full Deck?<abstract> <title style='display:none'>Abstract</title> <p>A standard deck of 52 playing cards has 4 suits: clubs (♣), spades (♠), hearts (♥) and diamonds (♦). Each suit has 13 denominations, or ranks, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (Jack), Q (Queen), K (King) and A (Ace). But sometimes, a card goes missing – lost between the couch cushions, clothes-pinned in the bike spokes, filched by the cat, eaten by the dog (see Figure 1), or maybe misplaced in a matching deck. You may find yourself not playing with a full deck. Building a house of cards? No problem. Playing poker? Problem. Or is it? Ask yourself the following question:</p> <p>What is the probability of being dealt a Two-Pair hand of five cards from a well-shuffled deck when the Ace of Spades is missing from the deck? Specifically, how does it compare to the probability of being dealt a Two-Pair hand from a standard 52-card deck?</p> <p>What does your ‘probability intuition’ tell you? On the one hand, with a missing Ace of Spades there are fewer Two-Pair hands. On the other hand, there are also fewer total five-card hands. The ratio of these two numbers defines the probability; which way it goes seems unclear. Let’s do the calculation – first for the standard 52-card deck and then for the 51-card deck missing the Ace of Spades.</p> </abstract>ARTICLEtrue the existence of decks that are not projective planes<abstract> <title style='display:none'>Abstract</title> <p>The game <italic>Spot It!</italic> is played with a deck of cards in which every pair of cards has exactly one matching symbol and the aim is to be the fastest at finding the match. It is known that finite projective planes correspond to decks in which every card contains <italic>n</italic> symbols and every symbol appears on <italic>n</italic> cards. In this paper we relax the hypothesis on the number of cards on which a symbol appears: we study symmetric decks in which every symbol appears the same number of times and we introduce the concept of <italic>maximal</italic> deck, providing a sufficient condition for a deck to have this property. We also produce various examples of interesting decks which do not correspond to projective planes.</p> </abstract>ARTICLEtrue One-Cushion Escape from Snooker in a Circular Table<abstract> <title style='display:none'>Abstract</title> <p>The one-cushion escape from snooker in a circular table can be viewed as a ge-ometric problem involving the reflection of a light ray in a circular mirror. There are at most four escapes in any given configuration. We will obtain specific configurations in which there are exactly 0, 1, 2, 3 or 4 escapes. The details consist in determining the number of real solutions in the interval (−1; 1) of certain polynomial equations, of degrees two, three or four.</p> </abstract>ARTICLEtrue Struggles of Chessland<abstract> <title style='display:none'>Abstract</title> <p>This is a fairy tale taking place in Chessland, located in the Bermuda triangle. The chess pieces survey their land and trap enemy pieces. Behind the story, there is fascinating mathematics on how to optimize surveying and trapping. The tale is written by the students in the PRIMES STEP junior group, who were in grades 6 through 9. The paper has a conclusion, written by the group’s mentor, Tanya Khovanova, explaining the students’ results in terms of graph theory.</p> </abstract>ARTICLEtrue remarks on the Game of Cycles<abstract> <title style='display:none'>Abstract</title> <p>The Game of Cycles is an impartial game on a planar graph that was introduced by Francis Su. In this short note we address some questions that have been raised on the game, and raise some further questions. We assume that the reader is familiar with basic notions from combinatorial game theory.</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 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<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 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 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 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 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 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 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 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 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 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