Solution:

Phoebe broke the machine and Monica is telling the truth.

> Rachel says Monica did it
If this is true, then the last statement must be true as well.

> Monica says Rachel is lying

> Phoebe says she did not.
If this is true, then one of the first two statements must be true as well.

Solution:

We now have a square of 3x3 size, 2 squares of 2x2 size and 5 squares of 1x1 size.

Solution:

Six.

There are only five colors, so six socks guarantee that two will be the same color.

Solution:

Never

Explanation:
In the question, the distance of x miles is given at a particular moment. The middle car is running at that time at 50 MPH. The distance will keep increasing with time as three are running at different speeds. It will increase by x miles every hour till a certain point of time and then it will start decreasing till the fastest car meets the slowest again.

As per question, all the cars stop then. Thus, the distance will never be x miles again.

Solution:

The check was for dollars 31.63.

The bank teller gave her dollars 63.31
She spent .05, and then she had dollars 63.26, which is twice the check.

Let x be the dollars of the check, and y be the cent.

The check was for 100x + y cent
He was given 100y + x cent

Also
100y + x - 5 = 2(100x + y)
Expanding this out and rearranging, we find:
98y = 199x + 5

or 199x ≡ -5 (mod 98)
or 98*2*x + 3x ≡ -5 (mod 98)
3x ≡ -5 ≡ 93 (mod 98)
this quickly leads to x = 31

Solution:

In the picture you can find the green matchsticks. We have moved them from the bottom and placed them in the middle. It was a bit tricky but now you have your desired result.

Solution:

3 1 2 2 1 1
1 3 1 1 2 2 2 1

Read Line as one 1 => 1
Read Line 2 as two one => 2 1
Read Line 3 as one two and one one => 1 2 1 1
and so on.

Solution:

Since the knife was in her hand which implies that she was planning to kill her husband "Inspector Montalbano".

Solution:

Slumdog Millionare = Slum + Dog + Million + air

Solution:

The password is U472BMT.

Read aloud the line and you will sound something like "U Four Seven 2 B M T".

Solution:

A Question