HOW MANY SQUARES AND RECTANGLES ARE DRAWN IN A SQUARE OR RECTANGLE DIVIDED INTO SMALL SQUARES?

Abstract

The aim of the research is to solve the problem: How many squares and rectangles are drawn in a square divided into n×n small squares or drawn in a rectangle divided into m×n (m<n) small squares? The researcher used induction to find public relations to solve the research questions, and he also used mathematical induction to prove the validity of the results. The researcher reached the following results:

1. The number of squares drawn in a square divided into n×n smaller squares is

n(n+1)(2n+1)/6

2. The number of rectangles (rectangular square) drawn in a square divided into n×n smaller squares equals

n^2(n+1)^2/4

3. The number of rectangles (their length differs from the width) drawn in a square divided into n×n smaller squares equals

n(n^2-1)(3n+2)/12

4. The number of squares drawn in a rectangle divided into m×n (m<n) small squares equals

m(m+1)(3n-m+1)/6

5. The number of rectangles (rectangular square) drawn in a rectangle divided into m×n (m<n) small squares equals

nm(n+1)(m+1)/4

6. The number of rectangles (their length differs from their width) drawn in a rectangle divided into m×n (m<n) small squares equals

m(m+1){3n(n-1)+2(m-1)}/12