Ever been asked to find the number of squares in a grid and thought it was difficult? Not anymore. Here is a simple formula that I propose, which can make things a lot easy.  

A grid has a large number of squares hidden in them. Have a look at this: 

  Counting the total number of all these types of squares can be a real difficult job.

  BG FORMULA is a simple way I have devised to calculate the number of squares in any grid.  

  If a grid has m rows and n columns, then the Total Number of Squares (TNoS) is given as;  

                                  For m<n;               TNoS =  Sm² + (n-m) Sm

NOTE: It is important that in any grid the lower number ( whether it is column or row ) is substituted in the position of m and the higher number is substituted in the position of n.

EXAMPLES

#1 The following grid has 2(m) columns and 2(n) rows.

  Hence the total number of squares present in the grid is = S2²+ (2-2) S2 = (1+4) + 0 = 5.

 You can count for yourself!! (The four smaller squares and the one big outer square)

#2 The following grid has 10(n) columns and 1(m) row.

Hence the total number of squares present in the grid is = S1² + (10-1) S1 = ( 1 ) + 9 = 10.

You can count for yourself again!!  (There are only 10 squares here and the other figures you can think of would be rectangles!!)

#3 The following grid has 10(n) columns and 7(m) rows.  

Hence the total number of squares present in the grid is = S7² + (10-7) S7= ( 1+4+9+16+25+36+49) + 3 * (1+2+3+4+5+6+7) = 224.

Try counting that!!. (First count all the smaller squares made of 1 block - 70. Then the squares made of 4 blocks, then squares made of 9 blocks and so on…)

PS: I DO NOT claim to be the FIRST or ONLY person to have discovered this formula. There maybe others who have proposed this formula before, but I am unaware of any such person as yet.

For further clarification e-mail me at blessongregory@gmail.com