Brain Teasers and Puzzles 

Workings

Hint
Think of squares and cubes.

Answer
512.

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

Workings

Hint

Answer
1680.

This is n to the power of 5, starting with 2^5 = 32, with the last digit missing, 7^5 = 16807.

Workings

Hint

Answer
293 031: these are the numbers 20, 21, 22, 23 etc grouped into three digits.

Workings

Hint
The sequence is related to 1, 2, 3, 4.

Answer
20.

Each term's initial letter position in the alphabet:

1 = One = O = 15th letter.
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4 = Four = F = 6th letter.
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20 = Ten = T = 20th letter.