Saturday, 20 October 2012

Color matching in concentric circles

A circle is divided into 8 equal sectors. Half are coloured red and half are coloured
blue. A smaller circle is also divided into 8 equal sectors, half coloured red and half
coloured blue. The smaller circle is placed concentrically on the larger. Prove that
no matter how the red and blue sectors are chosen it is always possible to rotate the
smaller circle so that at least 4 colour matches are obtained.

1 comment:

  1. define A_i for each sector of bigger circle as A_i=1 if the color in this sector matches the color of the sector of smaller circle overlapping it.
    0 o.w.
    N= number of overlapping sectors = sum of all A_i's
    give the smaller sector a random rotation. P(A_i=1)=1/2. E(A_i)=1/2.
    hence E(N)=8/2=4
    => there exists a particular rotation for which no of matching sectors is atleast 4