The corner poset with an application to an n-dimensional hypercube stacking puzzle

For any dimension n ≥ 3, we establish the corner poset, a natural triangular poset structure on the corners of 2-color hypercubes. We use this poset to study a problem motivated by a classical cube stacking puzzle posed by Percy MacMahon as well as Eric Cross’s more recent “Eight Blocks to Madness.” We say that a hypercube is 2-color when each of its facets has one of two colors. Given an arbitrary multiset of 2-color unit n-dimensional hypercubes, we investigate when it is possible to find a submultiset of 2n hypercubes that can be arranged into a larger hypercube of side length 2 with monochrome facets. Through a careful analysis of the poset and its properties, we construct interesting puzzles, find and enumerate solutions, and study the maximum size, S(n), for a puzzle that does not contain a solution. Further, we find bounds on S(n), showing that it grows as Θ(n2n)

Some combinatorial applications of Sage, an open source program

In this thesis, we consider the usefulness of Sage, an online and open-source program, in analyzing permutation puzzles such as the Rubik\u27s cube and a specific combinatorial structure called the projective plane. Many programs exist to expedite calculations in research and provide previously-unavailable solutions; some require purchase, while others, such as Sage, are available for free online. Sage is asked to handle a small permutation puzzle called Swap, and then we explore how it calculates solutions for a Rubik\u27s cube. We then discuss projective planes, Sage\u27s library of functions for dealing with projective planes, and how they relate to the card game Spot It! Since Sage is a free, open-source program, its limitations are a valid concern and are also discussed. --Abstract, page iii

Hamilton decompositions of 6-regular abelian Cayley graphs

In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made: Weak Lovász Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian. The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture: Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition. Alspach’s conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses. Chapters 1–3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspach’s conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators. Chapter 5 shows that if Γ = Cay(A, {s1, s2, s3}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ≤ i ≤ 3, Δi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Δi ≄ Cay(ℤ3, {1, 1}), then Γ has a Hamilton decomposition. Alternatively stated, if Γ = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Γ has a Hamilton decomposition if S has no involutions, and for some s ∈ S, Cay(A/(s), S) is 4-regular, and of order at least 4. Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups

Abstract Algebra: An Inquiry-Based Approach, Second Edition, Supplemental Material

This text is a supplement to Abstract Algebra: An Inquiry-Based Approach, Second Edition. It includes applications of algebra to RSA encryption, check digits, the games of NIM and the 15 Puzzle, and the determination of groups of small order. In addition, reference material that could be useful for some students appears in the appendices, such as background material on functions, methods of proof (including the equivalencies of the Well-Ordering Principle and different versions of mathematical induction), and complex roots of unity. The appendices also contain a complete proof that polynomial rings are rings, a proof of the Fundamental Theorem of Algebra, and a derivation of the formula for solving any cubic equation.https://scholarworks.gvsu.edu/books/1029/thumbnail.jp

Using Ideation Tools for Face-to-face Collaboration Within Complex Design Problems

The focus of this research are ideation tools and their ability to catalyse ideas to address complex design problems. Complex design problems change over time and the interactions among the components of the problem and the interaction between the problem and its environment are of such that the system as a whole cannot be fully understood simply by analyzing its components (Cilliers 1998, pp. I). Ideation for this research is defined as a process of generating, developing and communicating ideas that are critical to the design process (Broadbent, in Fowles 1979, pp. 15). Based on Karni and Arciszewski, who stated that ideation tools should act more like an observer or suggester rather than controller or an expert, I defne design ideation tools as tools or methods that enhance, increase and improve the user's ability to generate ideas with the client (Karni and Arciszewski 1997; Reineg and Briggs 2007). Based on a survey of over 70 ideation tools, protocol analysis of design activities, a web survey and semistructured interviews, I conclude that designers and clients may not have sufficient knowledge of ideation or ideation tools in either testing or practice as a catalyst for generating possibilities and that measuring ideation tools based on how many ideas they generate is misleading because it relates creativity and idea generation but does not adequately reflect the participants' experience. This research suggests that participants' cultural perceptions of design ideation and the design process actively inhibit idea generation and that a shift from design outcome led ideation tool design to designing ideation tools that engage design contexts are necessary to effectively address complex design problems. This research identifed a gap in ideation tools for designers to collaborate with their clients during the ideation phase to catalyse possibilities to complex design problems as the contribution to new knowledge

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum