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The 9 horizontal and 9 vertical lines on an 8X8 chessboard form 'r' rectangles& 's' squares. The ratio of s/r in the lowest terms is?

jee king king , 15 Years ago

Grade 12

2 Answers

Badiuddin askIITians.ismu Expert

Last Activity: 15 Years ago

Dear jee king

number of rectangles = r = ⁹C2 *⁹C2 =1296

number of square=s = (1² + 2² + 3² + ................+ 8² ) =204

so s/r =204/1296

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Badiuddin

Spandan Mallick

Last Activity: 6 Years ago

First for squares,

We see in a 8×8 chessboard, there are 64, 1×1 squares. Similarly 32, 2×2 squares and 16, 3×3 squares. Hence it forms a pattern. Also if you divide the area... total area is 64 sq. units.

For 1×1 squares, hence 64 are possible. For 2×2squares you can get it. Hence,

No of squares = (1²+2²+3²+...+8²) = 204.

For rectangles, consider the number of lines rather than squares. There are 9 lines. A rectangle can be formed by combination of any of these two four lines taking two from each pair of 9. Hence,

9C2 × 9C2 = 1296.

Spandan Mallick, IIT-JEE Aspirant