A bag contains n discs, made up of red and blue colours. Two
discs are removed from the bag. If the probability of selecting
two discs of the same colour is 1/2, what can you say about the
number of discs in the bag? Let there be r red discs, so P(RB) =
r/n (nr)/(n1), similarly, P(BR) = (nr)/n r/(n1). Therefore,
P(different) = 2r(nr)/(n(n1)) = 1/2. Giving the quadratic, 4r2
4nr + n2 n = 0. Solving, r = (nn)/2. If n is an odd square, n
will be odd, and similarly, when n is an even square, n will be
even. Hence their sum/difference will be even, and divisible by
2. In other words, n being a perfect square is both a sufficient
and necessary condition for r to be integer and the probability
of the discs being the same colour to be 1/2. Prove that
n(n+1)/2 (a triangle number), must be square, for the
probability of the discs being the same colour to be 3/4, and
find the smallest n for which this is true. What does this tell
us about n and n(n+1)/2 both being square? Can you prove this
result directly?