Random Puzzles 

Workings

Hint
You can't move diagonally.

Answer
5.

Reasoning
Starting with the central F, you can move:

Up (D), Up (I), Left (N).
Left (N), Left (I), Left (D).
Left (N), Left (I), Up (D).
Down (D), Left (N), Left (I).
Down (D), Down (I), Down (N).

Workings

Hint
3 raised to the 5^th power is 3 x 3 x 3 x 3 x 3.

Answer
3 raised to 5^th power.

Reasoning
3 raised to the 5^th power = 3 ^ 5 = 3 x 3 x 3 x 3 x 3 = 243

and

5 raised to the 3^rd power = 5 ^ 3 = 5 x 5 x 5 = 125.

Workings

Hint
Which painting won, and who came first?

Answer
#  Name     Artist
1  Alex     Monet
2  Emery    Constable
3  Glen     Van Gogh
4  Drew     Taylor
Reasoning
The four entrants were: Alex, Drew, Emery, Glen.
The four artists were: Constable, Monet, Taylor, Van Gogh.

By (2), the Monet came first.
   1           Monet
   2 
   3 
   4 

By (3), Glen came third.
   1           Monet
   2
   3  Glen
   4

By (1), the Constable can't have been last, so must have been second (as Emery painted it, and not Glen).
   1           Monet
   2           Constable
   3  Glen
   4

The Taylor was either third or fourth, either way, by (4) Alex must have won.
   1  Alex     Monet
   2           Constable
   3  Glen
   4

By (4), the Van Gogh came third, leaving Drew in last place.
   1  Alex     Monet
   2           Constable
   3  Glen     Van Gogh
   4  Drew

Leaving Emery in second place, and the Taylor in last place.
   1  Alex     Monet
   2  Emery    Constable
   3  Glen     Van Gogh
   4  Drew     Taylor

Workings

Hint
Fill the 9-litre container first.

Answer
Here are the contents of the containers after each stage:

A = 9 litres, B = 4 litres
A     B
9     0     fill A
5     4     fill B from A (leaving 5 litres in A)
5     0     empty B
1     4     fill B from A (leaving 1 litre in A)
1     0     empty B
0     1     fill B from A
9     1     fill A
6     4     fill B from A (only 3 more litres fit in B)
6     0     empty B

This leaves 6 litres in container A.