Random Puzzles 

Workings

Hint
I have fewer than 10 children.

Answer
4.

I have three daughters and one son.

Workings

Hint
Pages 1-9 are one digit each.

Answer
There are 65 pages and the number 9 appears six times.

Reasoning
The 65 pages are (with the six 9's highlighted):

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65.

If we remove the spaces, we get:

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465. Which is 121 digits.

Workings

Hint
The first letters of the words are: P, P, I, P, S, F, A, R, O, R.

Answers
----igy = prodigy ----aoh = pharaoh ----erg = iceberg ----ynx = pharynx ----lth = stealth ----mox = flummox ----ovy = anchovy ----esh = refresh ----awe = overawe ----ipt = receipt

Workings

Hint
There are no squares of size 5x5.

Answer
There are 125 squares without a rook.

Reasoning
There are 61 squares of size 1x1.
There are 39 squares of size 2x2.
There are 20 squares of size 3x3.
There are  5 squares of size 4x4.

Giving a total of 61 + 39 + 20 + 5 = 125 squares.