<abstract><title style='display:none'>Abstract</title> In this study, we define vedic cube as the layout of each digital root in a three-dimensional multiplication table. In order to discover the geometric patterns in vedic cube, we adopt two methods to analyze the digital root in a three-dimensional space. The first method is floor method, which divides vedic cube into several X-Y planes according to different Z values (floors) to analyze the geometric characteristics on each floor. The second method is symmetric plane method, which decomposes vedic cube by its main and secondary symmetric planes.</abstract>

Digital Root Patterns of Three-Dimensional Space

<abstract>
<title style='display:none'>Abstract</title>
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.
</abstract>

Flattening the Curve. . . of Spirographs

<abstract><title style='display:none'>Abstract</title>In this paper, we define the term <italic>Mathematical Sculpture</italic>, a task somehow complex. Also, we present a classification of mathematical sculptures as exhaustive and complete as possible. Our idea consists in establishing general groups for different branches of Mathematics, subdividing these groups according to the main mathematical concepts used in the sculpture design.</abstract>

A Classification of Mathematical Sculpture

<abstract><title style='display:none'>Abstract</title>Here I describe an infinite number of fractal tiles of regular rep-tiles in all dimensions above 1. Each rep-tile’s set of tiles can be divided into subsets based on certain visual characteristics. As fractals, they can be programmed and rendered in any size. They can be arranged in groups according to their aesthetic properties; used as an unending visual and pattern-recognition training ground for AI; and even animated as increments from one to the next.</abstract>

Infinite Tiles of Regular rep-tiles

<abstract>
<title style='display:none'>Abstract</title>
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>.
</abstract>

An alternative algorithm for the <italic>n</italic>–Queens puzzle

<abstract>
<title style='display:none'>Abstract</title>
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.
</abstract>

Go First Dice for Five Players and Beyond.

<abstract>
<title style='display:none'>Abstract</title>
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.
</abstract>

The One-Cushion Escape from Snooker in a Circular Table

<abstract>
<title style='display:none'>Abstract</title>
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.
</abstract>

The Struggles of Chessland

<abstract>
<title style='display:none'>Abstract</title>
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.
</abstract>

On the existence of <italic>Spot It!</italic> decks that are not projective planes

<abstract>
<title style='display:none'>Abstract</title>
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.
</abstract>

Fiboquadratic numbers and Rithmomachia

<abstract>
<title style='display:none'>Abstract</title>
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:
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?
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.
</abstract>

Not Playing With a Full Deck?

<abstract>
<title style='display:none'>Abstract</title>
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.
</abstract>

Some remarks on the Game of Cycles

<abstract><title style='display:none'>Abstract</title> Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.</abstract>

Independence and domination on shogiboard graphs

<abstract><title style='display:none'>Abstract</title> A Magic Cube of order p is a p×p×p cubical array with non-repeated entries from the set {1, 2, . . . , p3} such that all rows, columns, pillars and space diagonals have the same sum. In this paper, we show that a formula introduced in The Mathematical Gazette 84(2000), by M. Trenkler, for generating odd order magic cubes is a special case of a more general class of formulas. We derive sufficient conditions for the formulas in the new class to generate magic cubes, and we refer to the resulting class as regular magic cubes. We illustrate these ideas by deriving three new formulas that generate magic cubes of odd order that differ from each other and from the magic cubes generated with Trenkler’s rule.</abstract>

A generalization of Trenkler’s magic cubes formula

<abstract><title style='display:none'>Abstract</title> The classic n-queens problem asks for placements of just n mutually non-attacking queens on an n × n board. By adding enough pawns, we can arrange to fill roughly one-quarter of the board with mutually non-attacking queens. How many pawns do we need? We discuss that question for square boards as well as rectangular m × n boards. </abstract>

The Maximum Queens Problem with Pawns

<abstract><title style='display:none'>Abstract</title>A meaning of three dots (. . . ) and the Gauss’s sum.</abstract>

Yes, Gauss’s Answer is Indeed Correct!

<abstract><title style='display:none'>Abstract</title> Sudoku grids are often cited as being useful in cryptography as a key for some encryption process. Historically transporting keys over an alternate channel has been very difficult. This article describes how a Sudoku grid key can be secretly transported using quantum key distribution methods whereby partial grid (or puzzle) can be received and the full key can be recreated by solving the puzzle. </abstract>

Quantum Distribution of a Sudoku Key