12th Eötvös 1905 Problem 2 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 1/3 n . How many squares are uncolored? What is the total red area as n → ∞?
8 n uncolored squares after n steps. Total uncolored area = (8/9) n , so total red area = 1 - (8/9) n → 1 as n → ∞. 12th Eötvös 1905 © John Scholes jscholes@kalva.demon.co.uk 29 Oct 2003 Last corrected/updated 29 Oct 03