Random Puzzles 

Workings

Hint
How many different ways could the sign be placed back?

Answer
I simply pointed Chesterton back the way I had come and this left the sign in its correct orientation.

Workings

Hint
One of the words begins with the letter D.

Answer

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)

Workings

Hint
Take a close look at Emery's statements.

Answer
Drew.

Reasoning
Since each suspect told the truth just once, Emery's statements mean that it was either Drew or Alex.

If it was Alex, then both of Billie's statements would be true, which isn't allowed (and similarly both of Drew's would be false).

Therefore, it was Drew.

This can be double-checked with all of the other statements.