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.

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.

ReplyDelete0 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