rss_2.0Sciendo RSS Feed for Recreational Mathematics Magazinehttps://sciendo.com/journal/RMMhttps://www.sciendo.comhttps://sciendo-parsed.s3.eu-central-1.amazonaws.com/647368f74e662f30ba53d110/cover-image.jpghttps://sciendo.com/journal/RMM140216https://sciendo.com/article/10.2478/rmm-2024-0014ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00142024-09-20T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0010<abstract> <title style='display:none'>Abstract</title> <p><italic>A domino antimagic square of order n</italic> is an <italic>n</italic>×<italic>n</italic> array formed from a subset of the standard set of 28 dominoes such that the sums of the rows, columns, and two main diagonals form a set of 2<italic>n</italic> + 2 distinct, consecutive integers while an <italic>m</italic> × <italic>n domino antimagic rectangle is an m</italic> × <italic>n</italic> rectangular array formed from a subset of the standard set of 28 dominoes such that the sums of the rows and columns form a set of <italic>m</italic> + <italic>n</italic> distinct, consecutive integers. This paper outlines what the possible dimensions are for <italic>m</italic>×<italic>n</italic> domino antimagic rectangles and provides many examples of both domino antimagic rectangles and squares. Many open questions are given at the end of the paper for future exploration.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00102024-09-20T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0008<abstract> <title style='display:none'>Abstract</title> <p>Crossword puzzles lend themselves to mathematical inquiry. Several authors have already described the arrangement of crossword grids and associated combinatorics of answer numbers [Fer14] [Fer20] [McS16]. In this paper, we present a new graph-theoretic representation of crossword puzzle grid designs and describe the mathematical conditions placed on these graphs by well-known crossword construction conventions.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00082024-09-20T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0011<abstract> <title style='display:none'>Abstract</title> <p>In which we serve up several helpings of not- <italic>π</italic>.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00112024-09-20T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0012<abstract> <title style='display:none'>Abstract</title> <p>We discuss some open problems about occurrences of the digit one.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00122024-09-20T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0009<abstract> <title style='display:none'>Abstract</title> <p>In addition to being known for their other work, physicists George Gamow and Marvin Stern also published an analysis of an intriguing problem regarding their perception that although both worked on different floors, it seemed that the first elevator to arrive on their respective floors was traveling in the opposite direction of their destination. In this manuscript, a fictitious conversation occurs between the two in regards to the probability of the elevator stopping on every floor (which they term the Shabbat Effect). One claims that it is the number of elevator occupants that predominates whereas the other claims it is the number of floors in the building that predominates in giving rise to this effect. Equations for estimating the probability of the Shabbat Effect are developed and analyzed. The resulting equations show that the relationship between the number of elevator occupants and floors is surprisingly complicated. Consequently, no general rule can be given for whether the probability of the Shabbat Effect increases or decreases for an arbitrary change in the number of occupants and floors. However, it discovered that for similar changes in the number of occupants and floors, it is the number of floors that predominates in determining the probability of the elevator stopping on every floor.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00092024-09-20T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0013<abstract> <title style='display:none'>Abstract</title> <p>A mathematical model of Gregory Kavka’s “Toxin Puzzle” is constructed to provide the optimal course of action for this thought experiment. But the action suggested by the model is irrational, rendering the model useless for the rational player. Yet, it is the failure itself which provides a deeper insight into the nature of <italic>intention</italic> and mathematical modeling in general.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00132024-09-20T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0001<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00012024-03-08T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0002<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00022024-03-08T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0004<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00042024-03-08T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0003<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00032024-03-08T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0006<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00062024-03-08T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0005<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00052024-03-08T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2024-0007<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2024-00072024-03-08T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2022-0001<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2022-00012022-06-14T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2022-0005<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2022-00052022-06-14T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2022-0002<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2022-00022022-06-14T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2022-0003<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2022-00032022-06-14T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2022-0004<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2022-00042022-06-14T00:00:00.000+00:00https://sciendo.com/article/10.2478/rmm-2021-0010<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>ARTICLEtruehttps://sciendo.com/article/10.2478/rmm-2021-00102021-12-07T00:00:00.000+00:00en-us-1