Return to my Puzzle pages
Go to my home page

David Coles' Triangle Puzzle (Part I)

© Copyright 2001, Jim Loy

Part I - Introduction

3 lines, 1 triangle3,1: 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 lines, 2 triangles4,2: 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 lines, 4 triangles5,4: 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 lines, 7 triangles6,7: 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

This article is continued in David Coles' Triangle Puzzle (Part II). These diagrams were drawn using Cinderella and Paint Shop Pro.

Return to my Puzzle pages
Go to my home page