MathDB
Overlapping Squares

Source:

January 5, 2007
geometrynumber theoryrelatively prime

Problem Statement

Square S1S_{1} is 1×1.1\times 1. For i≥1,i\ge 1, the lengths of the sides of square Si+1S_{i+1} are half the lengths of the sides of square Si,S_{i}, two adjacent sides of square SiS_{i} are perpendicular bisectors of two adjacent sides of square Si+1,S_{i+1}, and the other two sides of square Si+1,S_{i+1}, are the perpendicular bisectors of two adjacent sides of square Si+2.S_{i+2}. The total area enclosed by at least one of S1,S2,S3,S4,S5S_{1}, S_{2}, S_{3}, S_{4}, S_{5} can be written in the form m/n,m/n, where mm and nn are relatively prime positive integers. Find m−n.m-n.
[asy] size(250); path p=rotate(45)*polygon(4); int i; for(i=0; i<5; i=i+1) { draw(shift(2-(1/2)^(i-1),0)*scale((1/2)^i)*p); } label("S1S_1", (0,-0.75)); label("S2S_2", (1,-0.75)); label("S3S_3", (3/2,-0.75)); label("⋯\cdots", (7/4, -3/4)); label("⋯\cdots", (2.25, 0));[/asy]
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