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What is the minimum number of non overlapping congruent triangles arranged in the plane, such that each vertex of the triangles coincide with exactly three triangles?

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    $\begingroup$ Can someone give me an example of this? $\endgroup$ – risky mysteries Jul 28 at 1:21
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    $\begingroup$ @risky In case you haven't seen it yet, check out Jaap's answer. $\endgroup$ – Daniel Mathias Jul 28 at 11:42
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The number of triangles in my best solution is

20

but I don't know if this is optimal.

If you take kites formed from four 30-30-120 triangles, you can place a number of them in a ring, like this:
enter image description here
This picture shows six such kites because that was easiest for me to draw, but you can remove one and make a loop with 5 kites, for a total of 20 triangles. @humn kindly made a picture of that:
enter image description here
If you use four kites then they start touching each other, so you'll get too many triangles at a vertex. @humn made a picture of that too:
enter image description here

Addendum:

I previously had an incorrect solution, as I used similar triangles instead of congruent ones (i.e. I used some triangles of a different size but the same shape). As requested by @humn, I'll keep that incorrect solution with 12 triangles available below:

The triangles are 20-80-80 isosceles triangles so that three can go together to form the corner of an equilateral triangle. There are then 3 smaller copies needed in the middle to fill things up and make sure there are three triangles at every vertex.
enter image description here

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  • $\begingroup$ These triangles are not congruent. They are only similar (same angles, different side lengths) $\endgroup$ – happystar Jul 28 at 7:43
  • $\begingroup$ @happystar I have changed my solution, so now it should be valid. $\endgroup$ – Jaap Scherphuis Jul 28 at 8:31
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    $\begingroup$ I think the four kite (16 triangles) solution is valid if the condition is taken to mean each vertex is a vertex of exactly three triangles. $\endgroup$ – Daniel Mathias Jul 28 at 11:40
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A liberal interpretation of “coincides” in the puzzle statement ...

“each vertex of the triangles coincides with exactly three triangles”

... allows for a vertex to touch a side, and not always another vertex, of another triangle as in these failed attempts ...


... that led to this pair of ...

... 16-triangle liberally-interpreted solutions.

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    $\begingroup$ Very nice solution, and in my opinion a completely valid one that improves on mine. $\endgroup$ – Jaap Scherphuis Aug 1 at 21:06
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suppose plan area is A and no of triangles are n,

A = n * 1/2 * h * b

so if we want n,

n = 2A /(hb), that is the minimum no of triangle

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