BrainDen.com - Brain Teasers
• 0

# Total number of equilateral triangles

## Question

*             *              *             *

*              *             *             *

*             *             *            *

*              *

*              *

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?

• 1

## Recommended Posts

• 0

41
The largest triangle that fits has sides of length 13.5.
Six others types in decreasing order have sides of length 2x3.5, 3, 7.5, 2, 3.5  and 1.
Of each there seem to be 1, 1, 2, 5, 7, 8 and 17, respectively.

Did I miss any?

##### Share on other sites

• 0

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

• 0

I get 41.

##### Share on other sites

• 0

I get 41.

But I cheeted:

```public class TriangleCount {

final int[][] p = {
{0, 0},   {2, 0},   {4, 0},   {6, 0},
{1, 1},   {3, 1},   {5, 1},   {7, 1},
{2, 2},   {4, 2},   {6, 2},   {8, 2},
{3, 3},   {5, 3},
{2, 4},   {4, 4},
};

void count() {
int count = 0;
for (int i = 0; i < p.length; i++)
for (int j = i + 1; j < p.length; j++)
for (int k = j + 1; k < p.length; k++) {
int a = (p[j] - p[i])*(p[j] - p[i]) + 3*(p[j] - p[i])*(p[j] - p[i]);
int b = (p[k] - p[j])*(p[k] - p[j]) + 3*(p[k] - p[j])*(p[k] - p[j]);
int c = (p[i] - p[k])*(p[i] - p[k]) + 3*(p[i] - p[k])*(p[i] - p[k]);
if (a == b && b == c) {
count++;
System.out.println("Equilateral triangle #" + count + ": i=" + i + " j="+j + " k=" + k);
}
}
}

public static void main(String[] args) {
new TriangleCount().count();
}
}
```

Edited by witzar
##### Share on other sites

• 0

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.)

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account. ×   Pasted as rich text.   Paste as plain text instead

Only 75 emoji are allowed.

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.