## David Coles' Triangle Puzzle (Part I)

Part I - Introduction

3,1: Dave Coles has invented a geometric puzzle. Draw a figure with n continuous lines, and produce the greatest possible number of triangles (with no lines or points inside them), and no other polygons. The smallest such object is a triangle made up of three lines (diagram on the left, imitating Coles' solution). We are also restrained to using Euclidean geometry. For example, these same three lines produce four triangles (three very large ones) on the surface of a sphere.

4,2: The second object is four lines, two triangles. It is possible to produce three polygons, using four lines (the second part of the diagram on the right). But one of these polygons must be a quadrilateral, which violates the rules. It may be easy to show that two triangles, and not three, are optimal. But you can already see that it will become difficult to determine the optimal solution for more complicated figures.

5,4: Here are two similar solutions (Mr. Coles' is on the left) of 5 lines, 4 triangles. Five lines can produce six polygons (a pentagram or 5-pointed star). Mr. Coles rates each solution with a score that is the ratio of triangles/lines. So for this one we have a score of 0.8. For optimal solutions, the score increases as the number of lines increases. I will mostly ignore the score here. But the score can help determine if a particular attempted solution is near optimal, or way off.

6,7: The first figure on the right is essentially Mr. Coles' solution (mirror image), except that he neglected to extend two of the lines, and so he missed a triangle. The second is a more regular figure, found while searching for an improvement.

Here is a table of the current records (1/19/01):

 lines triangles 3 1 4 2 5 4 6 7 7 10 8 14 9 18 10 22 11 27 12 32 13 38 14 44 15 50 16 54 17 60 18 72 19 76 20 84 21 92 22 110 23 114 24 122 25 130 26 156 27 160 30 210 32 224 34 272 36 288 38 342 40 320 42 420 44 440 46 506