Random Puzzles 

Workings

Hint
Look at the tops of the towers.

Answer
Add the two lines to the second tower from the left to complete the 'X' in the inverted letter sequence U, V, W, X, Y forming the towers' roofs.

Workings

Hint
One of the cheeses is Cheddar.

Answer
Edam, Cheddar, Brie, Cheshire.

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
Look at the number of sides of each of the shapes.

Answer
Square.

Reasoning
If we add the shape's sides, the total increases by 8, then 10, then 12, etc (2 more each time).
   triangle (3) + pentagon (5) + square (4) = 12

   square   (4) + hexagon (6) + hexagon (6) + square (4) = 20

   pentagon (5) + hexagon (6) + hexagon (6) + hexagon (6) + square (4) + triangle (3) = 30

Therefore:
   hexagon  (6) + octagon (8) + octagon (8) + octagon (8) + octagon (8) + square (4) = 42

Alternative Reasoning
Square the first shape's sides, and the remaining shapes add up to this number.
   triangle (3 x 3) = pentagon (5) + square (4)

   square   (4 x 4) = hexagon (6) + hexagon (6) + square (4)

   pentagon (5 x 5) = hexagon (6) + hexagon (6) + hexagon (6) + square (4) + triangle (3)

Therefore:
   hexagon  (6 x 6) = octagon (8) + octagon (8) + octagon (8) + octagon (8) + square (4)