Numerical Patterns and Geometrical Configurations

Abstract

Today, in contrast to Hilbert's assertion, most students have limited knowledge about configurations-except perhaps Pappus' 93, Desargues' 103 and Petersen's 103152 (the logo on the cover of the Journal of Graph Theory). These configurations are shown in FIGURE 1. The configuration labeled 93 illustrates Pappus' Theorem: If three points { 1, 3, 5 } on one line are joined in consecutive order to three points {4, 6, 2} on another line, the three intersection points {9, 7, 8} are collinear. The configuration 103 illustrates Desargues' Theorem: When extended, the corresponding sides of triangles (1, 2, 3) and (4, 5, 6) (which are said to be in perspective from the point 7), meet in a set of collinear points {8, 10, 9} (so that the triangles are said to be in perspective from a line also). In this note, we introduce and solve an interesting puzzle that serves to highlight the fascinating interplay between the geometric nature of point-and-line configurations and their representations as rectangular arrays of integers. We hope that our exposition will provide a rich source of challenges for students of varying levels of mathematical sophistication.