Arrowgrams: The Next Pencil and Paper Phenomenon

Abstract

Puzzles are a fascinating part of everyday life, and newspapers everywhere feature crosswords, word jumbles, and Sudokus as a way to test and intrigue the human mind. Arrowgrams are a new type of pencil and paper puzzle created by Dr. Kenneth Price of the University of Wisconsin Oshkosh. In these puzzles, directed graphs, which are a widely used physical element in both mathematics and computer science, are partially labeled, and the goal of the puzzle solver is to complete the rest of the labeling using the transitivity rule, which is a rule that resembles the Pythagorean Theorem. After completing the rest of the labeling, the solver may then find a secret message encoded in the puzzle. By analyzing arrowgrams using linear algebra, a puzzle creator can determine exactly what arrows need to be labeled for the puzzle to be uniquely solvable. A theorem is presented that relates which arrows need to be labeled for a special kind of directed graph called a tournament directed graph.