calculate the total no of squares of any size on a chess board

• Let us consider for the left hand column,
  for 1*1 square there are 8 squares, so 8 squares on the topmost row also,
  thus number of squares= 8^2 = 64
  
  for 2*2 squares there are 7 arrangements of squares on first column and also first row,thus no. of squares= 7^2 = 49
  
  for 3*3 squares we will get no. of squares= 6^2 = 36
  hence by this way we can say:
  
  Size Of square Number of squares
  --------------- -----------------
  1 * 1 8^2 = 64
  2 * 2 7^2 = 49
  3 * 3 6^2 = 36
  4 * 4 5^2 = 25
  5 * 5 4^2 = 16
  6 * 6 3^2 = 9
  7 * 7 2^2 = 4
  8 * 8 1^2 = 1
  ---------------
  Total no. of squares = 204
  We can also calculate this by using the formula= 1^2 +2^2 +3^2 +...+n^2
  = n(n+1)(2n+1)/6 where n=8
  = 204 (Ans)
  
• 12 years agoHelpfull: Yes(3) No(0)