### Find the missing number

See Answer### Answer with Explanation

Answer is 0 OR 16.

**How?**

Here, there is no special pattern except that they are continuous numbers which are jumbled up, so the missing number can either be 0 or 16.

Answer is 0 OR 16.

**How?**

Here, there is no special pattern except that they are continuous numbers which are jumbled up, so the missing number can either be 0 or 16.

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Answer is 9.

**How?**

Here, every middle element is the sum of the other two so answer should be 6+3 = 9.

**How?**

Lets count the number of triangles where side of the pentagon is one of the side of the triangle, for each one such side there are 6 triangles. but two of triangle has other side of the pentagon, those will be counted twice so we will count only one triangle for each side, so for five sides, total 5*5 = 25 such triangles.

Now, lets count the triangles where side of the pentagon is not the side of the triangle: there are 5+5 = 10 such triangles, this will be clear by the below diagram(5 green and 5 red triangles).

**How?**

Lets count…

By looking at it, first impression is … its very simple, But mind you! its not that easy !!!

So lets stick to the divide and conquer approach, we will divide this triangle into 3 equal triangles(triangle formed by connecting outer side to the center of the circle) and will count the number of triangles in each part and then by taking two or more parts together.

**Number of triangles in one part**: 4(non overlapping) + 3(overlapping) = **7 * 3 = 21**

**Number of triangles by taking two parts together**: 8 = **8 * 3 = 24**

**Number of triangles by taking all three parts together**: **2**

As all the sides are either horizontal or vertical we can say squares diagonals should be parallel. Now to see if the diagonals are parallel or not, i have drawn the diagonals where its not clear if it is a square or rectangle in the image below.

You can see i have labelled few sides as x and y to help you.

So,

Number of squares with side x/2 = 4

Number of squares with side x = 4

Number of squares with side y = 3

Number of squares with side x+y = 6

Number of squares with side x+y = 0

Number of squares with side x+2y = 1

Number of squares with side 2x+2y = 1

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Given

11×11 = (1+1)*(1+1) = 4

12×12 = (1+2)*(1+2) = 9

13×13 = (1+3)*(1+3) = 16

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Given 1 = 1^1 + 1,

4 = 4*4 + 4 = 20

6 = 6*6 + 6 = 42

9 = 9*9 + 9 = 90

10 = 10*10 + 10 = 110

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Given 1 = 5,

2 = 5*2 + (1) = 11

3 = 11*3 + (1 + 2) = 36

4 = 36*4 + (1 + 2 + 3) = 150

5 = 150*5 + (1 + 2 + 3 + 4) = 760

Looking at the symmetry, we can divide this triangle in 5 equal parts(square) and we will first count the number of triangles in each part and then number of triangles formed by combining 2 or more parts together.

**Number of triangles in one square**: 8(non overlapping triangles) + 8(overlapping triangles with middle lines a part of the triangle) =**16*5 = 80**

**Number of triangles by taking two squares(one middle and one other) at a time:** 4*4(there are four such combinations) = **16**

**Number of triangles by taking (only) three squares(one middle and two others) at a time:** 4(all squares should be a part of the triangle) *4(four such combinations) = **16**

**Number of triangles by taking 4 squares together: ** 2 *4(four such combinations) = **8**

**How?**

Lets count…

Looking at the symmetry, we can divide this triangle in 4 equal parts(square) and we will first count the number of triangles in each part and then number of triangles formed by combining 2 or more parts together.

**Number of triangles in one square**: 8(non overlapping triangles) + 4(overlapping triangles with middle lines a part of the triangle) =**12**

**Number of triangles by taking two squares at a time:** 6(try a little harder, you will fidn these) * 4(there are four such combinations) = **24**

**Number of triangles by taking (only) three squares at a time:** 1(all squares should be a part of the triangle) *4(four such combinations) = **4**

**Number of triangles by taking all squares together = 8**

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