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### Number Squares Chess Board Teaser

Number Squares Chess Board Teaser - 28 November

In the attached figure, you can see a chessboard and two rooks placed on the chess board. What you have to find is the number of squares that do not contain the rooks.

How many are there?

1. 124 or more

2. Interesting puzzle.
All squares of a chessboard are: Σ(n=0 to 8)n^2=204
That is 8*8+7*7+...+2*2+1*1=204.
The easy way to do it is to subtract the squares with a rook from each "class" of squares.
Squares of order 1. 1X1 = -2 (1 for each rook)
Sq. of order 2X2 = -8. 4 for each rook (imagine the rook at every square of 2X2=4 possible)
Sq. 3X3 =9 +9=18
Sq. 4X4 : 1st rook(at f3) =16-6 (down)=10
2nd rook (at d6)=16-6-1 (the square which contains the other rook)=9
Sq. 5X5 :from 16 total remains only 1 thus -15
Sq. 6X6, remains none --- -9
Sq. 7X7 none remains ---- -4
Sq. 8X8 none ---------------- -1
TOTAL REMAIN: 204-2-8-18-10-9-4-1= 152 squares that do not contain a rook.

3. There are 36 squares available for the other two rooks to be placed, so 34 squares don't have rooks.

4. The right answer ist 128
62 + (49-8) + (36-18) + (25-19) + 1