Q. How many squares are there in a 10X10 matrix.

• 385 squares of different sizes
  
  1×1 squares: 10x10 = 100
  2×2 squares: 9x9 = 81
  3x3 squares: 8x8 = 64
  .
  .
  .
  10×10 squares: = 1
  
  Total number of squares = 10^2 + 9^2+ 8^2 +...+1^2 = 385
• 13 years agoHelpfull: Yes(2) No(1)
• Total no of squares = 1^2+2^2+..+10^2= 385
• 13 years agoHelpfull: Yes(2) No(1)
• 385 squares of different sizes
  
  1×1 squares: 10x10 = 100
  2×2 squares: 9x9 = 81
  3x3 squares: 8x8 = 64
  .
  .
  .
  10×10 squares: = 1
  
  Total number of squares = 10^2 + 9^2+ 8^2 +...+1^2 = 385
• 13 years agoHelpfull: Yes(0) No(1)
• Answer is 385.
  
  Lets try to derive the General Pattern,
  
  1) n=1
  Matrix has only one number eg. 4.
  So, no. of squares for n = 1 is 1.
  2) n=2
  Matrix has four numbers.
  eg. 1 2
  2 4
  no. of 1x1 squares = 4.
  no. of 2x2 squares = 1.
  So, Total no. of Squares when n=2 is 5.
  3) n=3
  consider a 3x3 matrix of nine numbers.
  eg. 1 2 3
  2 4 6
  3 6 9
  no. of 1x1 squares = 9
  no. of 2x2 squares = 4
  no. of 3x3 squares = 1
  so, Total no. of squares for n=3 is 14
  Now, we compare the Results
  For n=1 ans is 1(1^2)
  for n=2 ans is 5(1^2+2^2)
  for n=3 ans is 14(1^2 + 2^2 + 3^2)
  ........................
  for n ans is the sum of 1^2 + 2^2 + 3^2 + ......+ n^2.
  ie sum of the squares of natural number = n(n+1)(2n+1)/6;
  
  Hence for n=10, we get no. of squares = 385
• 13 years agoHelpfull: Yes(0) No(0)
• total 385
  1*1
  2*2
  .
  .
  .
  .
  .10*10
  so total them we get 385
• 13 years agoHelpfull: Yes(0) No(0)
• for n*n matrix total no of square is given by 1^2+2^2+......+n^2
  in this case it is 1^2+2^2+......+10^2
  
  so by formula ie: [n*(n+1)*(2n+1)]/6
  
  => [10*11*21]/6
  
  => 385
  
  --------------------------------------------------
  
  NOTE: in n*n matrix total no of square are given by => [n*(n+1)*(2n+1)]/6
• 10 years agoHelpfull: Yes(0) No(0)