Divide the unit square into 9 equal squares (forming a 3 x 3 array) and color the central square red. Now subdivide each of the 8 uncolored squares into 9 equal squares and color each central square red. Repeat n times, so that the side length of the smallest squares is 1/3n. How many squares are uncolored? What is the total red area as n → ∞?
Solution
8n uncolored squares after n steps. Total uncolored area = (8/9)n, so total red area = 1 - (8/9)n → 1 as n → ∞.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03