Perfect triangles

This is the name for a family of triangles which all their sides are equal.

For instance: The basic 1-floor perfect triangle

This is a the simplest triangle to count.

The whole shape is a triangle so we have only 1 triangle in it.

Now let’s look at the 2-floors perfect triangle:

Let’s sort & count the triangles.

We can easily see there are 4 basic triangles in it. One is in the top floor and three are in the base floor. We can’t find any triangle which consist of 2 or 3 basic ones but…

The whole shape (which is made of 4 triangles) is also a triangle so we have 4+1 = 5 triangles in it.

And what about the 3-floors perfect triangle?

You can count by the floors from top downwards 1+3+5=9 basic triangles.

That means the floors number is doubled by itself.
If we examine the shape it is clear that there are no triangles that are made of 2 or 3 basic ones.

But when you look at the diagram you can see there are exactly 3 triangles that contain 4 basic ones.

There are no triangles which are made of 5 or 6 or 7 or 8 basic ones.

But of course there is one big triangle composed of 9 basic ones which is the whole shape.
Let’s sum it all up: we have a total of 9+3+1=13 triangles in this 3-floors regular triangle.

And how many triangles can you count in

the 4-floors perfect triangle?

And how many are in

the 5-floors perfect triangle?

…and in the 6-floors perfect triangle?

etc…

Can you find connections between the number of floors in such a triangle and the number of triangles you can find in it?

All these questions are answered in the e-book:

How to solve triangles counting riddles?

It will be available soon in Amazon.com!

One thought on “Perfect triangles”

1. The number of triangles were found from 1,2,1,2,1,2,1,2,1,2,… number sequence. The series of answers per n number of floors is the 3rd sum chain of the 1,2,1,2,1,2.,.
The first sum would be, 1,3,4,6,7,9,10,12,13,15,… 2nd sum chain is. 1,4,8,14,21,30,40,52,65,80,… and 3rd sum is 1,5,13,27,48,78,118,170,…which is the sum sequence from n=1 to any whole number.

Also, from this we can formulate a general formula of Se= (1/8)(n)(n+2)(2n+1) for even and So= Se-(1/8) for odd.