Puzzling the 120-cell

We introduce Quintessence: a family of burr puzzles based on the geometry and combinatorics of the 120-cell. We discuss the regular polytopes, their symmetries, the dodecahedron as an important special case, the three-sphere, and the quaternions. We then construct the 120-cell, giving an illustrated survey of its geometry and combinatorics. This done, we describe the pieces out of which Quintessence is made. The design of our puzzle pieces uses a drawing technique of Leonardo da Vinci; the paper ends with a catalogue of new puzzles.Comment: 25 pages, many figures. Exposition and figures improved throughout. This is the long version of the shorter published versio

The symbolic model for algebra : functions and mechanisms

The symbolic mode of reasoning in algebra, as it emerged during the sixteenth century, can be considered as a form of model-based reasoning. In this paper we will discuss the functions and mechanisms of this model and show how the model relates to its arithmetical basis. We will argue that the symbolic model was made possible by the epistemic justification of the basic operations of algebra as practiced within the abbaco tradition. We will also show that this form of model-based reasoning facilitated the expansion of the number concept from Renaissance interpretations of number to the full notion of algebraic numbers

Diffuse polarized emission associated with the Perseus cluster

We report on full-polarization radio observations of the Perseus cluster (Abell 426) using the Westerbork Synthesis Radio Telescope (WSRT) at wavelengths from 81-95 cm. We have employed a novel technique, Rotation Measure synthesis (Brentjens and de Bruyn, 2005) to unravel the polarization properties of the emission across the full field of view and detect polarized emission over a wide range of RM from about 0 to 90 rad m^-2. The low RM emission is associated with our Galaxy, while the high RM emission is associated with the Perseus cluster. The latter reaches typical surface brightness levels of 0.5-1 mJy per beam and must be rather highly polarized. Most of the peripheral polarized emission appears too bright, by about 1-2 orders of magnitude, to be explainable as Thomson scattered emission of the central radio source off the thermal electrons in the cluster. The bulk of the emission associated with the Perseus cluster is probably related to buoyant bubbles of relativistic plasma, probably relics from still active or now dormant AGN within the cluster. A lenticular shaped structure measuring 0.5-1 Mpc is strikingly similar to the structures predicted by Ensslin et al. (1998). At the western edge of the cluster, we detect very long, linear structures that may be related to shocks caused by infall of gas into the Perseus cluster.Comment: 18 pages, 17 figures, accepted by A&A, corrected small typo, added referenc

A Supplementary Topical Index

The following index has been designed to help the reader locate specific types of wordplay published in 26 issues of Word Ways from February 1978 through May 1984; it updates a similar index for 40 issues of Word Ways from February 1968 through November 1977 appearing in the February 1978 issue. Both indices use the same format: a logological core consisting of (1) letter-patterns in words, (2) operations upon letters in words, and (3) relationships between letters and sounds, and a periphery (the intersection of logology with other branches of wordplay) consisting of (1) literary wordplay and games, (2) academic language studies, and (3) word games and puzzles. Wordplay involving special sets of words (presidents, statenames, -cide words, etc.) is separately indexed

Carryless Arithmetic Mod 10

We investigate what arithmetic would look like if carry digits into other digit position were ignored, so that 9 + 4 = 3, 5 + 5 = 0, 9 X 4 = 6, 5 X 4 = 0, and so on. For example, the primes are now 21, 23, 25, 27, 29, 41, 43, 45, 47, ... .Comment: 7 pages. To the memory of Martin Gardner (October 21, 1914 -- May 22, 2010). Revised version (with a number of small improvements), July 7 201

The Simulation of a Computer-Assisted Instruction Program for Teaching a Non-Simulation Game: MEEMI-EQUATIONS Auto-Mate IMP Kit No. 1

LAYMAN ALLEN and Joan Ross have devised two extensions to the game of EQUATIONS. Allen has utilized the concept of mathematical balance in constructing games that bear a variety of academic content (for example, ON-WORDS and ON-SETS). The game quality in such cases depends on the complexity of problems that one player constructs for the others. The game is more durable when it is set in a metagame matrix that matches equally proficient players at each level of competence. When EQUATIONS is set in such a metagame it can become an Olympian struggle. In fact, during the past eight years there has been a National Academic Games Olympics in which students from many different states have participated. These Games have been held in Florida, Louisiana, Michigan, Ohio, and Pennsylvania. Schools interested in participating in future Games may secure information about them by writing to Robert W. Allen, Director; National Academic Olympics Project; Box 214; Newhall, California. The two games presented in Chapter 13 are additions to EQUATIONS to facilitate introduction of the basic game to beginners and to permit extension of the basic game to more sophisticated mathematical ideas for more experienced players. With EQUATIONS AUTO-MATE IMP KITS, Allen and Ross have arranged for a single individual to play by himself. In a series of look-up tables the other player is represented and his choices are indicated. Furthermore, the learning player\u27s mist,akes are explained and corrected. In effect, this variation transforms the basic structure of EQUATIONS into a teaching machine which retains the drama and uncertainty of plays of the actual games between two or more players. ADVENTUROUS EQUATIONS calls for several players and is similar to Dealer\u27s Choice in POKER. Each player can change the rules at the beginning of play as well as play the game. This change is an open invitation for the players to take part in the rule-making function and to experience control over the rule structure. Under the rules governing orderly change, both individual and group controls are exercised. The rules for change permit certain kinds of individual advantage, thus teaching that taking relevant and legitimate advantage is allowed. At the same time irrelevant advantages are banned as illegitimate. It is the teacher\u27s role to define relevant and legitimate changes. Since players will think of rules that the authors do not mention, this teacher function becomes a valuable dynamic tool in the mathematical and social education of the players. Be prepared for both delightful and sticky innovations. The authors will be interested in the innovations and your decisions. They are ready to serve as a court of appeal when needed. The senior author is a prolific game inventor who has concentrated on ways to practice academic content dynamically. He is interested in other game-inventor\u27s products, and will gladly advise and encourage them

The choice of toys by early childhood children

The article deals with the problem of choosing toys by children aged 0-48 months. It represents how children's situational involuntary interest in toys gradually transforms into cognitive interest. The latter determines the young children’s choices in various spheres of their life and, in particular, in choosing among the most accessible and feasible items – toys. The children’s cognitive interest in the choice of toys has been studied in the object and toy environment. A comparative analysis of children's choices among three types of the most common toys was carried out. It has been proved that figurative-interactive toys are the most popular among young children, figurative toys are in the second place, and collapsible toys are in the third place. The functional features of toys that encourage children to choose them have been revealed: sound, mobility, colour brightness. The more such features a toy has, the more often children choose it, the longer and more meaningful they play with it. It has been found that young children choose toy according to their psycho-physical, sensory capabilities and skills to act with it

The Perceived Impact of Morning Tubs on Elementary Students\u27 Twenty-First Century Skills

Increasing demands for creativity, critical thinking, problem-solving, and collaboration to be taught in schools puts pressure on educators to incorporate non-cognitive skills into the already rigid curriculum (Battelle for Kids, 2019). While there is ample research on play in the classroom and the importance of teaching twenty-first century skills to elementary students, there is very limited research on teaching twenty-first century skills via morning tubs. Morning tubs elicit skills such as collaboration, communication, creativity, motivation, critical thinking, and problem-solving in students (Partnership for 21st Century Skills, 2009). Therefore, this thesis aims to introduce play to educators as a strategy through which twenty-first century skills can be taught in the elementary education classroom. This qualitative study examines the implementation of morning tubs, an alternative to a morning worksheet, which consist of tactile objects designed to peak curiosity, creativity, and collaboration within students in a first-grade classroom in North Mississippi. The study took place during the beginning of the spring 2023 semester and consisted of semi-structured pre- and post- interviews with the researcher’s Clinical Instructor and four weeks of intervention during which the observer collected field notes on student behavior and interactions

RFID interactive tabletop application with tangible objects: exploratory study to observe young children’ behaviors

International audienceNumerous academic and industrial studies and developments concerning interactive tabletops are paving the way for new educational applications. We have developed an interactive tabletop application equipped with RFID technology. This tabletop, called TangiSense, is based on a Multi-Agent System that allows users to associate information with behaviors to manipulate tangible objects. The application involves the recognition of basic colors. With the application, children are required to manipulate tangible objects. Their task involves recognizing objects that have "lost" their dominant color and placing these objects in appropriate colored areas. A tangible magician object automatically analyzes the filled zones and provides children and their teacher with virtual and vocal feedback. This application has been evaluated in a field study with children 3 to 5 years of age. The initial results are promising and show that such an application can support interaction and collaboration, and subsequently educational situations, among young children

A course in mathematics appreciation

The purpose of this study is to create a mini-course in mathematics appreciation at the senior high school level. The mathematics appreciation course would be offered as an elective to students in the 11th or 12th grade, who are concurrently enrolled in trigonometry or calculus. The topics covered in the mathematics appreciation course include: systems of numeration, congruences, Diophantine equations, Fibonacci sequences, the golden section, imaginary numbers, the exponential function, pi, perfect numbers, numbers with shape, ciphers, magic squares, and root extraction techniques. In this study, the student is exposed to mathematical proofs, where appropriate, and is encouraged to create practice problems for other members of the class to solve. Also, areas for research are suggested so that the student may explore, even more deeply, areas which hold a particular interest for that student. These topics are treated with a three-pronged approach – historical, recreational, and practical. It is the author\u27s contention, supported by research, that this approach, along with the choice of topics, will assist in developing and enhancing the mathematics potential of the student to the highest possible extent