## 1.) How many triangles are in this picture ?

See Answer### There are a total of 20 triangles.

**How?**

Lets count…

Assuming smallest side of a triangle is 1 cms.

Number of triangles with side 1 cms: 12(1 + 5 + 5 +1)

Number of triangles with side 2 cms: 6(1 + 2(triangles facing upward), 1+2(triangles facing downward))

Number of triangles with side 3 cms: 2(1 triangle facing upward and 1 facing downwards).

**Thus, total number of triangles are 20.**

## 2.) How many triangles are in this picture ?

See Answer

### There are a total of 56 triangles in this picture.

**Puzzled How?**

Lets Count.

This is not as simple as the previous one, however we can solve it mathematically by using symmetry.

We can derive a formula as below

Total number of triangles(T) = 2*(A, Number of triangles in one big red triangle) + Number of triangles formed using both red parts(B);

Total number of triangles(T) = 2*(A, Number of triangles in one big red triangle) + Number of triangles formed using both red parts(B);

Number of triangles in one big red triangle can further be calculated as using the below formula.

**Number of triangles in one big red triangle(A) = 2*(C, Number of triangles in one green triangle) + number of triangles formed by combining both green parts(D).**

**Now, lets count the number of triangles in one green triangle**

**C is 9**.

**Now, lets count the number of triangles formed by combining both green triangles.**

**D is 7**.

As we know C and D we can calculate A, ** A= 2*C + D = 2*9 + 7 = 25**

**Now, lets count B(Number of triangles formed by using both red triangles together.**

**so B is 6**.

**Thus, Total number of triangles in the picture T = 2*A + B = 2*25 + 6 = 56.**

## 3.) How many triangles are in this picture ?

See Answer### There are a total of 28 triangles

**How?**

Lets count…

First we will count non-overlapping triangles.

Non-overlapping triangles between 1 & 2: 3

Non-overlapping triangles between 2 & 3: 4

Non-overlapping triangles between 3 & 4: 6

Non-overlapping triangles between 4 & 5: 4

Non-overlapping triangles between 5 & 6: 3

Total number of Non-overlapping triangles: 20

Now, Count overlapping triangles

**Thus, Total number of triangles in puzzle 3: 20+8 = 28**