Solution:

8 Steps

Explanation:
Steps------2L Gallon-------5L Gallon
A-------------12-------------0
B-------------07-------------5
C-------------07-------------0
D-------------02-------------5
E-------------02-------------0
F-------------00-------------2
G-------------12-------------2
H-------------09-------------5

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:

89

Explanation:
(1st-column-number * 2nd-column-number) + (1st-column-number + 2nd-column-number) = 3rd column number.

i.e

(3 x 1) + (3 + 1) = 3 + 4 => 7
(6 x 4) + (6 + 4) = 24 + 10 => 34
(8 x 7) + (8 + 7) = 56 + 15 => 71
(9 x 8) + (9 + 8) = 72 + 17 => 89

Solution:

FUCK

Explanation:
IF sound like the letter 'F'
YOU sound like the letter 'U',
SEE sound like the letter 'C',
KAY sound like the letter 'K'.

It's a clever way to use word fuck and it does not get blocked by filters.

Solution:

Tug of War

Solution:

3rd.

Explanation:
Rule: Take the bottom row, flip it, and put it on the top.

Solution:

20

Solution:

There are probably many, many solutions, five such solutions are:
> √(7^6) - 3^5
> (57 + 3)/.6
> ((3!)!- 5!)/(6 mod 7)
> floor(3*5*6.7)
> ceil(357 * sin (-6))

Solution:

569

Explanation:
Number in the triangle in ant-clockwise are: 15 12 and 18
The Central number is formed by there largest factor i.e
15=> 3 * 5 => largest factor is 5
12=> 2 * 6 => largest factor is 6
18=> 2 * 9 => largest factor is 9.

Therefore required number is 569.

Solution:

It can be done as shown below.