2023 Hang Lung Mathematics Awards Finalists’ Abstracts (listed by school name in alphabetical order) 1. Baptist Lui Ming Choi Secondary School Equilateral Polygons over Finite Fields The study of affine geometry over finite fields (as opposed to the rational, real or complex number fields) has been an interest among geometers. Most notably, the number of d-dimensional affine subspaces in a n-dimensional space over a finite field is a classic result dating back to the early 20th century, which involves the Gaussian binomial coefficient. Recently, N.J. Wildberger, in his book Divine Proportions, mentions the existence of a regular pentagon over a finite field of order 19. This has prompted a question: how many pentagons can be constructed over a finite field? More generally, how many equilateral polygons can be found, and how many are there up to symmetry? This paper investigates this problem, and gives both analytic and computational results on the number of equilateral polygons. 2. Diocesan Boys’ School Investigating the Edge 1-2 Conjecture: from Complete Graphs to Trees and Unicyclic Graphs The 1-2-3 Conjecture, posed by Karoński, Luczak, and Thomason, questions whether for every connected graph G different from K2, it is possible to assign a weight equal to 1, 2 or 3 to each edge such that for every two adjacent vertices in G, their sums of incident edge weights are distinct. Similar to the above problem, the Edge 1-2 Conjecture is to classify all graphs such that it is possible to assign a weight equal to 1 or 2 to each edge such that the above condition is satisfied. While the Edge 1-2 Conjecture is not true for all graphs G different from K2, for instance complete graphs, we will prove the Edge 1-2 Conjecture holds for all connected graphs that can be obtained by removing one edge from a complete graph, and classify all unicyclic graphs that can fulfil the Edge 1-2 Conjecture. Finally, we present a more efficient way of solving the 1-2-3 Conjecture for the specific case of triangle-union graphs, compared to the algorithm presented for the general 1-2-3 Conjecture.
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