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Packing equilateral triangles

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Keeping up with the triangle motif we've seen lately...

Suppose you have a regular hexagon. How many equilateral triangles can you pack in it if each vertex must touch an edge of the hexagon? What if the restriction was reduced to only needing to touch two edges of the hexagon? 

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Second question: Infinite...it doesn't state any size restrictions on the triangles...So something like this, ad infinitum:

triangles.thumb.png.360a067618787913ba84

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this doesnt appear to be infinite though 

I agree with Pickett :blush: you just need a stronger magnifying glass. 

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