# Viral math puzzle 1 : Proving 2 = 1 in 5 different methods

## 1) Using mathematical identity

a = b

a² = ab

a²-b² = ab-b²

(a+b)(a-b) = b(a-b)

cancel (a-b) from both side

so

a+b = b

2b = b

2= 1

So

where is the mistake? 🤔

### Solution

In the 5th step, we are cancelling (a-b)

That means we are dividing the equation with (a-b)

a = b so a-b = 0

That is we are dividing with zero, in mathematics division by zero is undefined

## 2) Using quadratic equation

Let’s consider *x²* + *x* + 1 =0. *x* ≠ 0. eq(1)

now we can divide both side by *x*

Then x+ 1 + 1/*x* = 0. eq(2)

From eq(1) we also have *x* + 1 = –*x²*, so we substitute this value into eq(2)

–*x²* + 1/*x* = 0……………..eq(2)

1/*x* = *x²*

1 = *x³*

*x* = 1, So we have *x* = 1, so let’s substitute in eq(1)

*x²* + *x* + 1 = 0

1²+ 1 + 1 = 0

3 = 0

we can subtract 1 in both sides then,

2 = -1

Take magnitude in both sides so,

| 2 | = | -1 |

Or, 2 = 1

Where is the mistake?🤔

## Solution

In the 2nd step, we are substituting *x* + 1 = –*x²* here we getting *-x² + 1/x = 0* this equation have solution x = 1 which is not a solution of * x² + x + 1 =0*

## 3) Using logarithm ( log2 )

Using log2 we have

log2 = 1 – ¹/₂ + ¹/₃ – ¹/₄ + ¹/₅ – ¹/₆ + ¹/₇ – ¹/₈ + ¹/₉ – …..

We can rearrange this series and we get

log2 = (1-¹/₂) – ¹/₄ + (1/3 – 1/6) – ¹/₈ + (¹/₅-¹/₁₀) –

¹/₁₂ + (¹/₇-¹/₁₄) – ¹/₁₆ + (¹/₉-¹/₁₈) – …..

After refusing above sum then we get

log2 = ¹/₂ – ¹/₄ + ¹/₆ – ¹/₈ + ¹/₁₀ – ¹/₁₂ + ¹/₁₄ – …….

= ¹/₂ × (1 – ¹/₂ + ¹/₃ – ¹/₄ + ¹/₅ – ¹/₆ + ¹/₇ – ……….)

log2= ¹/₂log2

1= ¹/₂

2 = 1

Where is the mistake?

## solution

In step 2 we rearrange the function,

log 2 is a non-converging function. We can’t rearrange a non-converging function

## 4) Using numerical method

Let’s take

-20 = -20

Then

16 – 36 = 25 – 45

4² – 4 × 9 = 5² – 5 × 9

Adding 81/4 both sides then we have

4² – 4×9 + (⁸¹/₄)= 5² – 5×9 + (⁸¹/₄)

Writing both sides in the form of expansion of (a + b)²

4² – 2 × 4 × (9/2) + (9/2)² = 5² – 2 × 5 × (9/2) + (9/2)

Both sides can be write like ( a+b )² form. so we get,

(4 – 9/2)² = (5 – 9/2)²

Take Square root in both sides

4 – 9/2 = 5 – 9/2

4 = 5

Subtract 3 from both sides

4 – 3 = 5 – 3

1 = 2

Or

2 = 1

So where is the mistake?

## Solution

We taking square root which is incorrect because

**√**(a²) = |a|

## 5) Using imaginary number ( *i* )

We know, 2 = 2

1 + 1 = 1 + 1

We can write 1 = √1

1 + 1 = 1 + √1

We can write, 1 = (-1)(-1)

1 + 1 = 1 + √((-1)(-1))

1 + 1 = 1+√(-1)√(-1)

Put √-1 = * i* , Then we get

1 + 1 = 1 + i×*i*

1 + 1 = 1 – 1

2 = 0

Divide both sides with 2 We get

1 = 0

Adding 1 in both sides Then we get

1+1 = 1

2 = 1

So where is the mistake?

## Solution

We taking square root which is incorrect because

**√**(a²) = |a| that is **√**1 = |-1| = 1

# Viral math puzzle 2 : Two answers and only one of them is the logically correct answer

There are two ways to solve the above problem

The first way to find the solution is to add the equation, then combine the sum with that of the previous equation

The second solution involves multiplying the second number of the equation by the number you are adding to it

The correct answer could either be 40 or 96.In the first case

For the second case

We have to get two solutions here this solution are getting viral on the internet but we can logically establish that 96 is true answer

In the first solution Summing two terms, then combine the sum with that of the previous equation.

Also, the number is in a pattern that is first 3 terms of the questions

So we can assume that 8+11 is not the 4th term and it is 8th term of the pattern so we can’t add 21 to 8th term.

If the pattern continued we have to get the same 96 in the second solution so 96 is the logically correct answer