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# Total number of equilateral triangles

Best Answer bonanova, 16 August 2014 - 01:20 PM

Go to the full post

7 replies to this topic

### #1 Perhaps check it again

Perhaps check it again

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Posted 15 August 2014 - 06:08 PM

*             *              *             *

*              *             *             *

*             *             *            *

*              *

*              *

The 16 points above lie in a plane on an equilateral triangular lattice.

Certain sets of three points of the figure correspond to the vertices of equilateral triangles.

Suppose you were to form all of the equilateral triangles possible, such that, for any given

equilateral triangle, it must have its three vertices coincide with three of the 16 points.

How many total equilateral triangles can be formed this way in the figure?

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### #2 bonanova

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Posted 16 August 2014 - 01:09 AM

Spoiler for Seems like

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### #3 Perhaps check it again

Perhaps check it again

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Posted 16 August 2014 - 03:08 AM

bonanova and other users,

there are additional equilateral triangles with side lengths different

from the correct (so far) three types you gave, and they have different orientations

from the ones you listed.

This problem is still open.

Edited by Perhaps check it again, 16 August 2014 - 03:09 AM.

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### #4 bonanova

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Posted 16 August 2014 - 01:20 PM   Best Answer

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### #5 bonanova

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Posted 04 September 2014 - 08:24 AM

Wondering whether a solution has been found.

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### #6 witzar

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Posted 04 September 2014 - 06:49 PM

Spoiler for

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### #7 witzar

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Posted 04 September 2014 - 07:15 PM

Spoiler for

But I cheeted:

Spoiler for

Edited by witzar, 04 September 2014 - 07:19 PM.

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### #8 bonanova

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Posted 04 September 2014 - 10:52 PM

Looks like that's the answer then. Nice.

I'm wondering whether that's the optimal configuration for 16 points to generate ETs.

I'm also wondering whether the number of ETs increases proportionately faster then the number of points.

It's easy to see that adding a 17th point at 6,4 increases the ratio from 2.5625 to 2.70588 (46 ETs.)

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