The Mystic Rose

six-pointed mystic rose

@standupmaths pointed me at a puzzle by @sciencepunk at MathsJam: or at the Science Punk blog.

The question is: how many triangles are there in a 'Mystic Rose' shape like this one (right) - with six interconnected points.

I reckon there are 110.

I got this by splitting it up into different types of triangle, like this:

Triangle type Number Notes
One triangle 18 6 on the edge (isosceles), 12 at the corners (right-angled)
2T 18 12 x edge + corner (right-angled); 6 x double-corner (equilateral)
3T 12 Right-angled triangle along edge
4T 6 All the way along the edge (isosceles)
1 triangle, one quad 12 Right-angled triangle involving vertex and centre
3T 1Q 6 Equilateral triangle into middle
2T 2Q 6 Centre and two non-adjacent corners (isosceles)
2T 3Q 6 Isosceles triangle with two opposite corners
3T 3Q 12 Right-angled triangle with non-adj corners
5T 3Q 12 Right-angled triangle with three vertices
6T 6Q 2 "Star of David" equilateral triangles
Total: 110

I keep adding it up differently, but I'm pretty happy that it's 110 now.

Note that all of the triangles involve at least one vertex. Apart from the "star of David" triangles in the middle, the isosceles/equilateral triangles come in groups of six and the right-angled (scalene) triangles come in groups of 12.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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2 comments on “The Mystic Rose

  • Frank the SciencePunk

    Good work! Unfortunately the feature of all triangles having a vertex doesn’t last – larger values for n give rise to triangles buried in the rose that don’t touch the edge of the shape.

    The real challenge though, is “how many triangles does a rose of n sides hold?”

    • Colin

      Thanks, Frank!

      Yeah, when I extracted my compasses and drew out the seven-sided one, I noticed the ones in the middle. Really good puzzle, thanks for sharing it!

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