# Most triangles formed by three triangles

What is the maximum number of triangles you can form by drawing three triangles on a piece of paper?

Good luck!

• Do all the sides have to be drawn or can we use the sides of the paper? Is the paper flat? – chasly - supports Monica Jul 30 at 13:55
• all sides need to be drawn and the paper is flat (as usual). – Dmitry Kamenetsky Jul 30 at 14:29
• I tried to find a proven result about this, but it's very hard to figure out search keywords. The number of results on "maximum triangles" is astonishing. :) – Ross Presser Jul 31 at 20:18

I've managed to get

30 triangles

As follows

The trick here is

To make the three triangles maximally intersecting. I'm not sure if this is optimal here though.

Counting

Note here that each red kite region as shown yields two large triangles and two small triangles. Going around the outer vertices we see that the large triangles are distinct so there are 18 of them but the small triangles are each counted twice so there are 9 of them and then we add the three original triangles to get 30. • Nice answer. This could be our winner... – Dmitry Kamenetsky Jul 30 at 14:30
• This seems to be the maximum, ran python script on 100 000 random configurations, no more than 30. – Alexey Burdin Jul 31 at 16:50
• ... or else you need to reshape the non-triangle shapes in the diagram (the nonagon and quadrialterals) to be triangles. That part seems impossible ... still working on it. – Ross Presser Jul 31 at 19:35
• Playground: geogebra.org/geometry/sv7kqwbn – Ross Presser Jul 31 at 20:01
• – Alexey Burdin Jul 31 at 22:12

This might not be optimal, but its the best one I could find:

It has 18 triangles.

here are some of the triangles highlighted for clarity

and then, of course, the 3 original triangles.

Every larger distinct triangle has a side with a different colour and a number adjacent to that triangle. The smaller distinct triangles have a number inside them, except in cases where the triangle is very small, in which case the number is adjacent to the triangle. The three overlapping triangles generate twenty-four distinct triangles. • The diagram seems identical to hexomimo answer, but the triangle count is different. – Dmitry Kamenetsky Aug 1 at 0:43