Hi, this is Presh Talwalkar.
How many triangles do you see in this picture?
Give this problem a try, and when your ready, keep watching the video for the solution.
We need to count the number of triangles very carefully.
We'll first consider triangles which have a size of 4, which means it's side length is equal to 4 times the smallest triangle.
There is exactly 1 triangle of this type, and this triangle is facing upward.
Let's continue counting only the number of triangles which are facing upward.
Let's count the number of triangles which have size 3.
In the bottom row, there are 2 of these triangles, and 1 row above that there's 1.
So that makes for a total of 3 triangles of size 3 facing up.
How many triangles are there size 2 facing up?
The bottom row has 3, the next row has 2, and 1 row above that has 1.
3 + 2 + 1 is 6.
Finally, we count the number of triangles of size 1 facing up.
The bottom row has 4, the next row has 3, the next row has 2, and the top row has 1.
4 + 3 + 2 + 1 is equal to 10.
Now we need to count the number of triangles which are facing down.
We'll start by counting the number of triangles of size 1.
The bottom row has 3, the next row has 2, and the row above that has 1.
3 + 2 + 1 is equal to 6.
Now, we count the number of triangles of size 2 facing down, and there is exactly 1.
Now, as you can see, this is the largest size triangle that's facing down.
So there are no triangles of size 3 or 4 facing down.
So we have now listed out all of the triangles
and we can total this up, to get that there are 27 triangles in this picture.
Now, what would happen if we want to generalize this,
For a base that has 𝘯 smaller triangles?
Well we could- We ha- Error.
We found a pattern in the shape we just saw.
The number of triangles facing up of size 𝘯,
(the largest size), was equal to one
The next size, there were 2 more of them
The next size there were more of them, according to the shape.
Each time we were adding the next number in the sequence
These are the triangular numbers which are the shapes
You can form that have a triangular arrangement of shapes
and they're equal to the sum of the numbers from one to 𝘯.
The triangles of the smallest size, 1, were equal to the triangular number of size 𝘯 .
Now in many videos, we've figured out the way, to sum up the numbers from one to 𝘯
and this is the triangular formula number,
which is equal to 𝘯(𝘯+1)/2
to count the number of triangles facing down
we start from the triangles of size 1
This will be offset by 1
so we start out with triangular number of 𝘯-1
to get to the next size of 2, we have to offset it by 2
because each time the row is decreasing in size.
So this is going to go about halfway up
and then the largest two sizes theney the largest size, we're gonna have 0 triangles facing down
so now we've figured out a pattern,
in how to count the number of triangles of each size
and we have a formula for the number of triangular numbers
equal to 𝘯(𝘯+1)/2
so this actually going to vary whether you have n equal
𝘯 is going to be even or 𝘯 is going to be odd
but there's a nice way you can summarize this if you add up all these numbers
the total number of triangles is
going to be n(n+2) (2n+1) all over 8, rounded down.
this will give you the number of triangles, it's actually a sequence
The number of triangles formed if you have this equilateral grid of triangles the
sequence starts out 1 5 13 27 was what we had when we had four triangles in our
grid it continues 48 78 and so on.
did you figure it out?
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