Université de l’Alberta, 24 - 27 juin 2016
Through our work we are able to generate larger calcrostics, now of size 5x5 and 7x7 instead of 3x3, more complicated ones, now involving rational numbers rather than integers, and puzzles that require many more conditions to be satisfied through extra diagonal relations.
Apart from many new procedures that had to be written for handling these new types of puzzles, the key problem was that much larger systems of under-determined polynomial equations needed to be solved with computer algebra. We developed a method of computing special solutions of highly under-determined and highly non-linear, polynomial equations. The new technique was implemented and merged into the computer algebra package CRACK. It was successfully used for the generation of all types of new calcrostic puzzles.
i) We formalize and prove "the most general setting" to define associativity of a binary function and generalized associativity, i.e. "inserting parentheses in any manner".
ii) By definition, the order of an element is either a positive integer or infinite. Using transfinite recursion, we generalize the notion to the class of ordinals for topological groups. Arithmetical and group-theoretic properties of such generalization are studied. We also discuss examples that lead to further questions.