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## David Coles' Triangle Puzzle (Part I)

© Copyright 2001, Jim Loy

** 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 |

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

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