There is a simple and elegant solution for the case of 3 people
(i.e. part 1), which doesn't require much formal math. However, this
is not the case for more than 3 people. The solution requires using
the error correct coding theory, specifically, the distance-3 Hamming
code.
However, there is a simple and elegant proof for the upper bound
without using much formal math, i.e. it's impossible to come up with a
strategy with winning probability > 1 - 1 / 2^k.