@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.

## 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!