Random Puzzles 

Workings

Hint
Try reversing the numbers.

Answer
891.

If we reverse each of the numbers we get:

   203 · 202 · 201 · 200 · 199 · ?

Which is counting down from 203, and the next number is 198, which is 891 when reversed.

Workings

Hint
Look at the start of each word.

Answer
augment.

The first three letters are the same as the first three letters of the months in order.

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.

Workings

Hint
Write the numbers without any spaces.

Answer
02.

These are the numbers 10 11 12 13 14 15 16 17 18 19 20 21 22, but spaced incorrectly.