In
many geometric puzzles they ask about no. of Squares or Rectangles in a Square
or a Rectangle. Essentially these questions are pertaining to Permutations and
Combinations. However these questions can be solved by simple mathematical
formulae.
(1)
No of squares in a square of size nxn: Sum of
squares of n natural numbers.
={ n(n+1)(2n+1)}/6
(2)
No. of Rectangles in a Square of size nxn: Sum
of squares of n natural numbers.
={ {n(n+1)}/2]2
(3)
No. squares in a rectangle of size mxn= mxn +
(m-1)(n-1) + (m-2)(n-2) +……+ 0
(4)
No. Rectangles in a Rectangle of Size mxn
= ( sum of natural nos from 1 to m) ( sum of natural nos from 1 to n)
= {m(m+1)/2}{n(n+1)/2}
(5)
No. of Triangles in a Triangle= {n(n+2)(2n+1)}/8
; where n is the no. of triangles at the base of the larger Triangle. This
formula is to be used when n is even.
(6) No. of Triangles in a Triangle= {n(n+2)(2n+1) - 1}/8 ; where n is the no. of triangles at the base of the larger Triangle. This formula is to be used when n is odd.
(6) No. of Triangles in a Triangle= {n(n+2)(2n+1) - 1}/8 ; where n is the no. of triangles at the base of the larger Triangle. This formula is to be used when n is odd.