Sherlock Holmes was challenged to make a pre-drawn line smaller without erasing it. He did it in fractions of seconds. How?

I am thinking of a five-digit number such that:
The first and last digits are the same, their submission is an even number and multiplication is an odd number and is equal to the fourth number. Subtract five from it and we obtain the second number. Then divide into exact halves and we get the 3rd number.

What number I am thinking of?

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

A man has to get a fox, a chicken, and a sack of corn across a river.

He has a rowboat, and it can only carry him and one other thing.
If the fox and the chicken are left together, the fox will eat the chicken.
If the chicken and the corn are left together, the chicken will eat the corn.

How does the man do it?

Can you identify the Oscar-winning movie from the rebus below?

Take away the whole and some still remain. What is it?

I don't get anymore wetter, no matter how much rain falls on me.

The dirtier I get, the whiter I look. Who am I?

In the given figure, you can see that four match sticks are used to form a square. Can you form five squares by using six matches?

I am a seven-letter word but if you remove four of my letters I will have one left. What word am I?