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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?

In this paper we discuss a natural generalization of Stern-Brocot sequences which comes from the introduction of weighted mediants. We focus our attention on the case k = 3 , in which 2 a + c 2 b + d and a + 2 c b + 2 d are the two mediants inserted between a b and c d . Our main result is a determination of which rational numbers between the starting terms appear in the union of the Stern-Brocot sequences.

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?”

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.

A magic SET square is a 3 by 3 table of SET cards such that each row, column, diagonal, and anti-diagonal is a set. We allow the following transformations of the square: shuffling features, shuffling values within the features, rotations and reflections of the square. Under these transformations, there are 21 types of magic SET squares. We calculate the number of squares of each type. In addition, we discuss a game of SET tic-tac-toe., Comment: 22 pages, 27 figures, 15 tables

We discuss some old common knowledge puzzles and introduce a lot of new common knowledge puzzles.

In this paper, we discuss coin-weighing problems that use a 5-way scale which has five different possible outcomes: MUCH LESS, LESS, EQUAL, MORE, and MUCH MORE. The 5-way scale provides more information than the regular 3-way scale. We study the problem of finding two fake coins from a pile of identically looking coins in a minimal number of weighings using a 5-way scale. We discuss similarities and differences between the 5-way and 3-way scale. We introduce a strategy for a 5-way scale that can find both counterfeit coins among 2k coins in k + 1 weighings, which is better than any strategy for a 3-way scale.

Consider equipping an alphabet $\mathcal{A}$ with a group action that partitions the set of words into equivalence classes which we call patterns. We answer standard questions for the Penney's game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions., Comment: 32 pages, 1 figure

A group of students in 7-9 grades are inventing combinatorial impartial games. The games are played on graphs, piles, and grids. We found winning positions, optimal strategies, and other interesting facts about the games., Comment: 12 pages, 1 figure

In this paper we study halving-edges graphs corresponding to a set of halving lines. Particularly, we study the vertex degrees, path, cycles and cliques of such graphs. In doing so, we study a vertex-partition of said graph called chains which are equipped with interesting properties.

Kasha-eating dragons introduce advanced mathematics. The goal of this paper is twofold: to entertain people who know advanced mathematics and inspire people who don’t. Suppose a four-armed dragon is sitting on every face of a cube. Each dragon has a bowl of kasha in front of him. Dragons are very greedy, so instead of eating their own kasha, they try to steal kasha from their neighbors. Every minute every dragon extends four arms to the four neighboring faces on the cube and tries to get the kasha from the bowls there. As four arms are fighting for every bowl of kasha, each arm manages to steal one-fourth of what is in the bowl. Thus each dragon steals one-fourth of the kasha of each of his neighbors, while at the same time all of his own kasha is stolen. Given the initial amounts of kasha in every bowl, what is the asymptotic behavior of the amounts of kasha? Why do these dragons eat kasha? Kasha (buckwheat porridge) is very healthy. But for mathematicians, kasha represents a continuous entity. You can view the amount of kasha in a bowl as a real number. Another common food that works for this purpose is soup, but liquid soup is difficult to steal with your bare hands. We do not want to s soup spilled all over our cube, do we? If kasha seems too exotic, you can imagine less exotic and less healthy mashed potatoes. How does this relate to advanced mathematics? For starters, it relates to linear algebra [3]. We can consider the amounts of kasha as six real numbers, as there are six bowls, one on each of the six faces of the cube. We can view this six-tuple that represents kasha at each moment as a vector in a six-dimensional vector space of possible amounts of kasha. To be able to view the amounts of kasha as a vector, we need to make a leap of faith and assume that negative amounts of kasha are possible. I just hope that if my readers have enough imagination to envision six four-armed dragons on the faces of the cube, then they can also imagine negative kasha. The bowl with 2 pounds of kasha means that if you put two pounds of kasha into this bowl, it becomes empty. For those who wonder why dragons would fight for negative kasha, this is how mathematics works. We make unrealistic assumptions, solve the problem, and then hope that the solution translates to reality anyway.

We investigate a type of a Sudoku variant called Sudo-Kurve, which allows bent rows and columns, and develop a new, yet equivalent, variant we call a Sudo-Cube. We examine the total number of distinct solution grids for this type with or without symmetry. We study other mathematical aspects of this puzzle along with the minimum number of clues needed and the number of ways to place individual symbols., Comment: 24 pages, 17 figures

We research a combinatorial game based on the Cookie Monster problem called the Cookie Monster game that generalizes the games of Nim and Wythoff. We also propose several combinatorial games that are in between the Cookie Monster game and Nim. We discuss properties of P-positions of all of these games. Each section consists of two parts. The first part is a story presented from the Cookie Monster's point of view, the second part is a more abstract discussion of the same ideas by the authors., Comment: 18 pages, 1 figure

This chapter begins with the following teaser about hungry dragons stealing kasha from one another. “Suppose a four-armed dragon is sitting on every face of a cube. Each dragon has a bowl of kasha in front of him. Dragons are very greedy, so instead of eating their own kasha, they try to steal kasha from their neighbors. Each minute, every dragon extends four arms to the four neighboring faces on the cube and tries to get the kasha from the bowls there. As four arms are fighting for every bowl of kasha, each arm manages to steal one-fourth of what is in the bowl. Thus, each dragon steals one-fourth of the kasha of each of his neighbors, while at the same time all of his own kasha is stolen. Given the initial amounts of kasha in every bowl, what is the asymptotic behavior of the amounts of kasha?” It then arrives at a branch of mathematics called representation theory.

Semi-bar k -visibility graphs are graphs for which vertices can be drawn as horizontal segments with left endpoints on the y -axis (semi-bars) and edges can be drawn as vertical segments (sightlines) so that two semi-bars are visible to each other if and only if there is a sightline which intersects them and at most k other semi-bars. Cyclic semi-bar k -visibility graphs are graphs for which vertices can be drawn as segments with one endpoint at the origin (semi-bars) and edges can be drawn as arcs of circles centered at the origin (sightlines) so that two semi-bars are visible to each other if and only if there is a sightline which intersects them and at most k other semi-bars. We show that every semi-bar or cyclic semi-bar k -visibility graph can be represented in the plane with vertices drawn as points in convex position and edges drawn as segments so that there are no k + 2 pairwise crossing edges. Furthermore, we prove that the class of graphs having cyclic semi-bar k -visibility representations with semi-bars of different lengths is the same as the class of ( 2 k + 2 ) -degenerate graphs that can be represented in the plane with vertices drawn as points in convex position and edges drawn as segments so that there are no k + 2 pairwise crossing edges and so that adding any edge would result in k + 2 pairwise crossing edges.

For n >= 2 a construction is given for a large family of compact convex sets K and L in n-dimensional Euclidean space such that the orthogonal projection L_u onto the subspace u^\perp contains a translate of the corresponding projection K_u for every direction u, while the volumes of K and L satisfy V_n(K) > V_n(L). It is subsequently shown that, if the orthogonal projection L_u onto the subspace u^\perp contains a translate of K_u for every direction u, then the set (n/(n-1))L contains a translate of K. If follows that V_n(K), 19 pages, 3 figures

If the number of ones is 0, then the only non-zero, non-first digit of such a number is 2. This 2 should be included in the autobiography. Since the third digit of the number is not zero, it must be 2. Therefore, the number has 2 twos, and so must be 2020. Similar arguments for the case of 1 one and 2 ones give us the full set of autobiographical numbers: 1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000. A short piece by Fred Gavin (Mathematics Magazine, vol 66:4,1993) also gives the deriva tion.

This chapter examines the problem of the “Cookie Monster number.” In 2002, Cookie Monster® appeared in the book The Inquisitive Problem Solver by Vaderlind, Guy, and Larson, where the hungry monster wants to empty a set of jars filled with various numbers of cookies. The Cookie Monster number is the minimum number of moves Cookie Monster must use to empty all the jars. The chapter analyzes this problem by first introducing known general algorithms and known bounds for the Cookie Monster number. It then explicitly finds the Cookie Monster number for jars containing cookies in the Fibonacci, Tribonacci, n-nacci, and Super-n-nacci sequences. The chapter also constructs sequences of k jars such that their Cookie Monster numbers are asymptotically rk, where r is any real number, 0 ≤ r ≤ 1.

This chapter presents solutions to generalizations of Konstantin Knop's puzzle for any number of minutes and any number of parallel scales. It first describes the similarity of the original puzzle with a multiple-pans problem: a coin-weighing puzzle involving balance scales with not two, but any number of pans. The notion of a coin's potential is defined next. The chapter then provides a solution to the parallel weighing problem in case there is an unlimited supply of real coins. Afterward, the chapter offers a solution to the original puzzle and its generalization for any number of minutes, and generalizes these results to the use of more than two scales in parallel. In addition, the find-and-label variation of this problem for any number of minutes is discussed, before the chapter concludes with a comparison of the find-and-label problem with the just-find problem.

In this paper, we consider a modular extension to the game of Nim, which we call $m$-Modular Nim, and explore its optimal strategy. In $m$-Modular Nim, a player can either make a standard Nim move or remove a multiple of $m$ tokens in total. We develop a winning strategy for all $m$ with $2$ heaps and for odd $m$ with any number of heaps., 17 pages

A new variation of an old hat puzzle, where sages are standing in line one behind the other.