Le Monde puzzle [#1121]

A combinatoric puzzle as Le weekly Monde current mathematical puzzle:

A class of 75<n<100 students is divided at random into two groups of sizes a and b=n-a, respectively, such that the probability that two particular students Ji-ae and Jung-ah have a probability of exactly 1/2 to be in the same group. Find a and n.

(with an original wording mentioning an independent allocation to the group!). Since the probability to be in the same group (under a simple uniform partition distribution) is

\frac{a(1-1)}{n(n-1)}+\frac{b(b-1)}{n(n-1)}

it is sufficient to seek by exhaustion values of (a,b) such that this ratio is equal to ½. The only solution within the right range is then (36,45) (up to the symmetric pair). This can be also found by seeking integer solutions to the second degree polynomial equation, namely

b^\star=\left[ 1+2a\pm\sqrt{1+8a}\right]/2 \in \mathbb N

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.