كم عدد المربعات و المستطيلات المرسومة في مربع أو مستطيل مقسم إلى مربعات صغيرة ؟

Abstract EN

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

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

The number of rectangles (rectangular square) drawn in a rectangle divided into m×n (m

The number of rectangles (their length differs from their width) drawn in a rectangle divided into m×n (m