Volume 8 (2021): Edizione 14 (October 2021)

If water is flowing at the same constant rate through each of H ⩾3 hoses, so that any one hose will fill any one of J ⩾ 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.

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.

In this paper a new method for solving the problem of placing n queens on a n×n 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 n Queens completion puzzle. Furthermore, this method allows finding repetitive patterns of solutions for any n.

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.

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.