#2           62. DEATH BY TRIANGULATION                  
“Yes, it was indded sad to hear of the death of our colleague, Peter de Ferrug. A tragic loss. You may be interested in this sheet I found amongst his papers.”

I took the proffered sheet. At the top were some line diagrams (shown left). Underneath there was a table:

Number of lines
3
4
5
6
7
Number of non-overlapping triangles
1
2
5
7
11

There was no diagram for the seventh case, but a small note at the bottom of the page read: I have assuredly found an admirable figure of seven lines, but the space is too narrow to contain it.

Can you supply the diagram?
 Hints and strategies 

Hint 1                     Solution                      Extensions
HINT 1

You’ll just have to play with this one, looking for sets of lines which maximize the number of triangles.
HINT 2

You should have obtained remainders 6, 3, 7 for the respective departments. Also remainders 3, 2, 2, 2, 5 for F (Foster), K, P, R and W; all other names have remainder 0. You should now immediately be able to place five people in their departments.
SOLUTION
EXTENSION

1. Mathematicians will recognize Pierre de Fermat in this story. Look him up in your library or on the internet. Look for the story about the note he left in a margin. For example, you might look at

http://en.wikipedia.org/wiki/Pierre_de_Fermat

2. There are a number of problems which involve the drawing of lines in the plane. See if you can create some of them.