Brain Teasers and Puzzles 

Workings

Hint
How many 1 x 1 squares are there?

Answer
There are 116 squares without a rook.

Reasoning
There are 61 squares of size 1 x 1.
There are 37 squares of size 2 x 2.
There are 15 squares of size 3 x 3.
There are  3 squares of size 4 x 4.

Giving a total of 61 + 37 + 15 + 3 = 116 squares.

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.

Workings

Hint
How many 2 x 2 squares are there?

Answer
There are 128 squares without a rook.

Reasoning
There are 62 squares of size 1 x 1.
There are 41 squares of size 2 x 2.
There are 18 squares of size 3 x 3.
There are  6 squares of size 4 x 4.
There is   1 square  of size 5 x 5.

Giving a total of 62 + 41 + 18 + 6 + 1 = 128 squares.

Workings

Hint
There are no larger squares than size 4x4.

Answer
There are 110 squares without a rook.

Reasoning
There are 60 squares of size 1x1.
There are 35 squares of size 2x2.
There are 12 squares of size 3x3.
There are  3 squares of size 4x4.

Giving a total of 60 + 35 + 12 + 3 = 110 squares.

Workings

Hint
How many 8x8 squares are there, what about 7x7.

Answers
204 squares. 1,296 rectangles, which also includes the squares.

Workings

Hint
Try this with a real board.

Answer
5 queens are required.

Workings

Hint
How many 2x2 squares are there?

Answer
There are 93 squares without a rook.

Reasoning
There are 60 squares of size 1x1.
There are 33 squares of size 2x2.

Giving a total of 60 + 33 = 93 squares.