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Doodling With the Stars

We all like to draw stars with five lines...but, at its basis is an interesting problem.

Question: How many disjoint triangles can be created with n lines?

Now, when n = 1 or n = 2, no triangles can be created...ala the definition of a triangle.

When n = 3, you can make a triangle. Note: you can assume that the lines end at vertices or cross to form vertices.

So, what about n = 4 or n = 5 or n = 6?


So, our sequence K(n) for n = 1, 2, 3, 4, 5, 6, ... becomes 0, 0, 1, 2, 5, 7...

Is there a pattern? Can you find K(7)? Or can you find the function value for K(n) for any n?

Please explore this problem before more information is shared (i.e. next week, this problem will include some commentary).

 


Hint: Draw some pictures using 7 lines...then 8 lines...

Look at the previous examples...Is there a best approach for drawing the lines to determine K(n) for n lines?

 


Solution Commentary: You should discover that K(7) = 11, K(8) = 15, and K(9) = 20. But what is K(10)?

Unfortunately, this remains an unsolved problem in geometry (or number theory)? According to Erich Friedman (Stetson University), at best we know K(10) is either 25 or 26, K(11) is either 32 or 33, and K(12) is 48, 49, or 50.

Wow...and the odd thing, is that we know K(13) = 47 and K(15) = 65...but at best, K(14) is 53, 54, or 55.

Isn't mathematics fascinating...as to what is known and what is unknown!