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vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pmFollowed by: 4 members; Thanked: 29 times

by vipulgoyal » Tue Mar 19, 2013 3:23 am

A LARGE SQUARE IS EQUALLY DIVIDED INTO 16 SMALLER SQUARES. HOW MANY RECTANGLES AND SQUARES DOES THE LARGE SQUARE CONTAIN.?

Ans 100 & 30

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Source: Beat The GMAT — Problem Solving | Tue Mar 19, 2013 3:23 am

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Anju@Gurome: GMAT Instructor; Posts: 511; Joined: Wed Aug 11, 2010 9:47 amFollowed by: 86 members; Location: Delhi, India; Thanked: 344 times

by Anju@Gurome » Tue Mar 19, 2013 3:39 am

vipulgoyal wrote:A LARGE SQUARE IS EQUALLY DIVIDED INTO 16 SMALLER SQUARES. HOW MANY RECTANGLES AND SQUARES DOES THE LARGE SQUARE CONTAIN?

I'll post a detailed analytical solution later. Meanwhile, if you are good with formulas, here you go...

Number of squares in a nxn chess board = 1Â² + 2Â² + 3Â² + ... + nÂ² = n(n + 1)(2n + 1)/6
Number of rectangles in a nxn chess board = [n(n + 1)/2]Â²

Here, n = 4
Hence, number of squares = 1Â² + 2Â² + 3Â² + 4Â² = 1 + 4 + 9 + 16 = 30
Hence, number of rectangles = [4(4 + 1)/2]Â² = [20/2]Â² = 100

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Anju@Gurome: GMAT Instructor; Posts: 511; Joined: Wed Aug 11, 2010 9:47 amFollowed by: 86 members; Location: Delhi, India; Thanked: 344 times

by Anju@Gurome » Tue Mar 19, 2013 4:14 am

First I will describe the number of rectangles as it is bit easier than the number of squares.

In an nxn chessboard, there are (n + 1) vertical lines and (n + 1) horizontal lines.
To form a rectangle, we have to choose 2 of the vertical lines and 2 of the horizontal lines.

Hence, number of rectangles = (Number of ways to select 2 vertical lines from (n + 1) vertical lines)*(Number of ways to select 2 horizontal lines from (n + 1) horizontal lines) = [(n + 1)C2]*[(n + 1)C2] = [(n + 1)C2]Â² = [n(n + 1)/2]Â²

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vipulgoyal: Master | Next Rank: 500 Posts; Posts: 468; Joined: Mon Jul 25, 2011 10:20 pmFollowed by: 4 members; Thanked: 29 times

by vipulgoyal » Tue Mar 19, 2013 4:26 am

HI Anju got it for rectangle, how did u get formula for squre ??

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Anju@Gurome: GMAT Instructor; Posts: 511; Joined: Wed Aug 11, 2010 9:47 amFollowed by: 86 members; Location: Delhi, India; Thanked: 344 times

by Anju@Gurome » Tue Mar 19, 2013 4:27 am

If a large square is equally divided into 16 smaller squares, we will get a figure like this...

For any square of size 1 x 1, the left vertical edge can be in any one of the 4 positions. Similarly, the top horizontal edge can occupy any one of the 4 positions. So the total number of 1x1 squares = 4*4 = 16

For a 2x2 square, the left vertical edge can occupy any of the 3 positions and the top horizontal edge also any of the 3 positions, giving 3*3 = 9 squares of size 2x2.

Continuing in this way we get 2*2 = 4 squares of size 3x3 and 1 square of size 4x4.
Hence, total number of squares = 1 + 4 + 9 + 16 = 30

Generalizing this for nxn chessboard,

Number of 1x1 squares = n*n = nÂ²
Number of 2x2 squares = (n - 1)*(n - 1) = (n - 1)Â²
...
Number of (n - 2)x(n - 2) squares = 3*3 = 3Â²
Number of (n - 1)x(n - 1) squares = 2*2 = 2Â²
Number of nxn squares = 1*1 = 1Â²

Hence, total number of squares = 1Â² + 2Â² + 3Â² + ... + nÂ²

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