What has to be broken before you can use it?

You are given a cube that is made with the help of 10x10x10 smaller cubes summing up to a total of 1000 smaller cubes. You are asked to take off one layer of the cubes.
How many remain now?

If eleven plus two equals one, what does nine plus five equal?

Our product and our sum always give the same answer. Who are we?

If two's company, and three's a crowd, then whats four and five ?

If you’ve got me, you want to share me; if you share me, you haven’t kept me. What am I?

If you remove one from eleven, it becomes ten. If you remove one from nine, it becomes ten.

How is this possible?

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

Only one color, but not one size,
Stuck at the bottom, yet easily flies.
Present in sun, but not in rain,
Doing no harm, and feeling no pain.
What is it?

You need to move three matchsticks to form three squares. Can you do it?