The 9 horizontal and 9 vertical lines on an 8 × 8 chessboard form ‘r’

Solution:

Choosing any 2 verticles and 2 horizontal lines would give us a rectangle.

Note: few, in this case, will also be squares. We can count them as rectangles only.

For number of Rectangles \(r\):

Choosing 2 vertical lines and 2 horizontal line \(= 9C2 \times 9C2 = 36 \times 36 = 1296\)

For number of squares \(s\):

If we see there will be \(8\times 8 = 64\) 1*1 squares.
There will be \(7 \times 7 = 49\) 2*2 squares.
and so on..

So number of squares (s) \(= 8^2+7^2+6^2....1^2 \)(sum of n squares) \(= \frac{n(n+1)(2n+1)}{6} = \frac{8\times 9 \times 17}{6} = 204\)

So Ratio \(\frac{s}{r} = \frac{204}{1296} = \frac{17}{108}\) (in lowest terms)

Hence the right answer is Option A.

Note: if in \(r\) only and only rectabgles are considered then the answer will be \(= \frac{204}{1296-204} = \frac{204}{1092}\)

Bunuel please share detailed answer. Thanks in advance

Every square is a rectangle. So essentially, the question is similar to asking the following: out of every possible rectangle that can be made on the 8 by 8 board, how many are squares?



(1st) total number of rectangles (including squares)

If you think about a rectangle drawn on the coordinate plane, you can identify the rectangle by:

2 distinct X coordinates

And

2 distinct Y coordinates.


For instance, a square/rectangle drawn from the origin that has length of 4 and is located in quadrant 1 will be:

(0.0) —-(4.0) ——(0.4) —-(4.4)


Letting the 2 X coordinates be A and B
and
Letting the 2 Y coordinates be P and Q

(A , P) —- (B , P) —— (A, Q) —- (B , Q)

Thus, if you can picture the 8 by 8 chessboard on the coordinate plane, we would be able to identify any unique rectangle by:

Picking 2 unique horizontal lines out of the 9 drawn

And

Picking 2 unique vertical lines out of the 9 drawn

Total rectangles = “9 choose 2” * “9 choose 2” = 36 * 36


(2nd) how many of these total rectangles are squares

1st we can easily find the 1 by 1 squares - they will be the number or squares on the 8 by 8 chess board ——-> (8)^2


2nd, to figure out how many 2 by 2 square exist - you can count from the first row up - there will be 7 unique squares in the 1st column, and there will be 7 columns more

———> (7)^2

This pattern will continue all the way until you get to the amount of 8 by 8 squares = (1)^2

So we are looking for the sum of the first 8 consecutive perfect squares which is given by:

= (1/6) (2n + 1) n (n + 1) =

= (1/6) (2*8 + 1) * 8 * (8 + 1)

=. (1/6) (17) (8) (9)

= 17 * 4 * 3

17 * 12


Final answer:

(17 * 12) / (36 * 36)

Cancel the factor of 12 in the NUM and the DEN

(17) / (3 * 36) =

17/108

Answer is B not A?

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