Random Puzzles 

Workings

Hint
Try removing the spaces between the numbers.

Answer
48 and 11.

Reasoning
These are the squares of numbers without the correct spacing.

   1 4 9 16 25 36 49 64 81 100

with no spaces becomes:

   149162536496481100

broken into blocks of two:

   14 91 62 53 64 96 48 11 00

So the next two in the sequence are 48 and 11.

Workings

Hint
You can only go up, or right.

Answer
6.

With U = Up, and R = Right, you can move in the following ways:
UURR
URUR
URRU
RUUR
RURU
RRUU

Workings

Hint
The code has just four letters, A, B, C, and D. Each letter encodes 5 different letters.

Answer
I think, therefore I am.

Reasoning
A represents a, b, c, d, and e.
B represents f, g, h, i, and j.
C represents k, l, m, n, and o.
D represents p, q, r, s, and t.
E represents u, v, w, x, and y.

Workings

Hint
Think of squares and cubes.

Answer
512.

Each term alternates between a square and a cube, 8 3 = 512.