Medium Puzzles 

Workings

Hint
This is a well known nursery rhyme.

Answer
HDD.

The initial letters from the nursery rhyme, Hickory Dickory Dock.
Hickory, dickory, dock,
The mouse ran up the clock,
The clock struck one,
The mouse ran down,
Hickory, dickory, dock.

Workings

Hint
Try this with a real board.

Answer
5 queens are required.

Workings

Hint
Think of the phrase 'my bucket and spade'.

Answer
In the phrase 'my bucket and your spade', the gap between 'my' and 'and' is the same as the gap between 'and' and 'spade', namely one word.

Indeed, this logic can be continued to give the following sentence. In the first sentence the gap between 'and and' and 'and and' is seven words, et cetera.

Workings

Hint
Say each number, one digit at a time.

Answer
11131221141113122112.

Reasoning
Each term describes the previous term.

For example, 42 = one four, one two = 1412.

And, 1412 = one one, one four, one one, one two = 11141112.