The Mystic Petals Puzzle
The Mystic Petals Puzzle was first posed to me by Frank Swain in the pub after some science event we had been to. It was easy to grasp, and seemed straight forward, but has since defied being solved. Having reduced me to a whimpering mathematical heap, I now pass it on to you.
It is an extension of the Mystic Rose puzzle, which is where you place a number of equally spaced points on an exact circle and then join each point to all of the others with straight lines.
The basic challenge is: for a given number of points "n", how do you calculate how many lines you will need to draw? There is a fantastic investigation into this puzzle – with an interactive applet – on the nrich website:
I recommend having a go at solving this problem first.
Frank's ingenious idea was to try and count how many triangles are possible for each value of n. Each triangle needs to be drawn using the lines made by the Mystic Rose pattern, or sections of these lines.
This starts off nicely for n=3 with just the one possible triangle.
For n=4 you now have two different types of triangles, with four of each. Eight triangles in total.
Once you hit n=5 the triangle start to hot-up a bit. There are now seven different types, with five of each of them. But it's still fairly straight forward to count thirty-five triangles in total.
Now all you need to do is continue this pattern and work out the relationship between the number of points and the number of triangles!
If you do solve it yourself, please let us know:
Matt Parker matt@standupmaths.com
Please note that although several people have worked on this problem, none of us have looked it up to see if it is already solved. Finishing a maths puzzle that someone else has already done is still just as exciting if you don't know the answer. Only once we all give up completely in a decade or two will we look it up. So don't just email us someone else's solution.
You can read about Frank's adventures with the Mystic Petals here: http://scienceblogs.com/sciencepunk/2010/06/counting_the_petals_of_mystic.php
This page is totally copyright Matt Parker 2010.