## Of Hexagons and Triangles

October 13, 2012 by xavierjoseph

This week’s NPR Weekend Edition Sunday puzzle challenge went like this:

“Draw a regular hexagon, and connect every pair of vertices except one. The pair you don’t connect are not on opposite sides of the hexagon, but along a shorter diagonal. How many triangles of any size are in this figure?”

I suspect a mathematical formula exists to determine the answer. But as I don’t know of such a formula, nor do I know how to find one, I resorted to the manual method of counting.

This turned out to be not as simple as it sounds. I spent the better part of the week counting and re-counting and counting again, coming up with many different answers. Of course, there is only one answer, hence my determination to keep counting until I found it!

I began by drawing a hexagon and randomly connecting vertices, leaving just one pair unconnected (depicted here circled in red):

Then I began to look for triangles of any size. The first time I counted, I found 50. Figuring there could easily be some that I missed, I counted again more carefully and found 76. The third count yielded 84.

I then took a more careful and systematic approach to counting. It occurred to me that there is a symmetry in the figure that can be readily seen if the hexagon is rotated slightly:

Notice now that the left and right halves are mirror images. Knowing this made counting easier because if I found a triangle on one half of the image, that meant a corresponding triangle was on the opposite half of the image, making it less likely that I’d miss counting a triangle.

Starting with triangles comprised of once piece, I carefully labeled those triangles with a ‘1’. Next, I looked for triangles composed of two pieces and labeled those triangles with a ‘2’. I continued this method up to ‘10’, which is the greatest number of pieces I could find that formed a triangle. Adding up the subtotals for each number of pieces came to 80 triangles.

I went over all of the labeled images again and found some that I had mislabeled and some that I had missed. This time, adding the subtotals yielded 82 triangles. After double- and triple-checking my labels and arithmetic, I settled on 82 as my final answer:

[click image to enlarge]

On tomorrow’s Weekend Edition Sunday program, I’ll find out whether I counted correctly!

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on November 20, 2012 at 8:44 pm |Daniel J. Caldwell (@dcaldwell1)If you put back the missing line, I count 110.

on November 21, 2012 at 9:01 pm |xavierjosephCorrect! It seems odd to me that the number of triangles, when all lines are included, is 110 because 110 is not divisible by 3, 4, 6 or 8.