Random Puzzles 

Workings

Hint
Try it once, and then try it again.

Answer
You get 202 each time!

Workings

Hint
Try writing the numbers out as letters.

Answer
45.

Reasoning
Each term increases by the number of letters in the previous term (ignoring any hyphens).
1  + length("ONE")       = 4
4  + length("FOUR")      = 8
8  + length("EIGHT")     = 13
13 + length("THIRTEEN")  = 21
21 + length("TWENTYONE") = 30
30 + length("THIRTY")    = 36
36 + length("THIRTYSIX") = 45

Workings

Hint
This is a very common sequence.

Answer
X.

The last letters of the words one, two, three, four, five, six.

Workings

Hint
A little algebra might help.

Answer
¹/₈.

Reasoning
If we convert the question to algebra, we have:
((n^2 + 4) x 2 – 3) x 4 – n = 20

Expanding the brackets and simplifying gives:

(2n^2 + 8 – 3) x 4 – n = 20

    (2n^2 + 5) x 4 – n = 20

         8n^2 + 20 – n = 20

              8n^2 – n = 0 (*)

                8n – 1 = 0

                    8n = 1

                     n = ¹/₈

In the equation marked (*) zero is also a potential solution, but as the question tells us that we "Start with a number larger than 0" we know that n can't be 0, and therefore we can safely divide by n.

Double-Checking
                 ¹/₈
   squared     = ¹/₆₄
   add 4       = 4¹/₆₄
   double      = 8¹/₃₂
   take away 3 = 5¹/₃₂
   times 4     = 20¹/₈

   subtracting ¹/₈ gives 20, as required.