Counting

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 = 1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6
  Number of rectangles in a nxn chess board = [n(n + 1)/2]²

Here, n = 4
Hence, number of squares = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
Hence, number of rectangles = [4(4 + 1)/2]² = [20/2]² = 100

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]² = [n(n + 1)/2]²

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 = n²
  Number of 2x2 squares = (n - 1)*(n - 1) = (n - 1)²
  ...
  Number of (n - 2)x(n - 2) squares = 3*3 = 3²
  Number of (n - 1)x(n - 1) squares = 2*2 = 2²
  Number of nxn squares = 1*1 = 1²

Hence, total number of squares = 1² + 2² + 3² + ... + n²