Find the mistake in the below maths equations

A = 2
A(A-1) = 2(A-1)
A2-A = 2A-2
A2-2A = A-2
A(A-2) = A-2
A = 1

I am the one who remains awake when you sleep. But I need to rest when you awake. Who Am I?

I am a word of 12 alphabets.

* Alphabets 12, 4, 7, 2, 5 : Eastern beast of burden
* Alphabets 1, 8, 10, 9 are street made famous by Sinclair Lewis.
* Alphabets 11, 3, 6 are past.

On whole it means a person suffering from delusions of greatness.

What Is it ?

Find The Odd One Out

1. FLOW
2. SNIP
3. TRAP
4. DRAW
5. BACK

I am dry when I go inside, but wet when I am pulled outside.
The longer I am inside, the stronger I get
When I am completely out, I am wet and I literally sag
Who am I?

Find the next number in the series

12, 20, 40, 72, 116, 172 ?

A bomb exploded in a room full of married couples and no one died. How?

Can you find a number that lies one third of the distance between 1/3 and 2/3?

Always in you, Sometimes on you;

If I surround you, I can kill you.

What am I?

15 caves are arranged in a circle at the temple of doom. One of these caves has the treasure of gems and wealth. Each day the treasure keepers can move the treasure to an adjacent cave or can keep it in the same cave. Every day two treasure seekers visit the place and have enough time to enter any two caves of their choice.

How do the treasure seekers ensure that they find the treasure in the minimum number of possible days?