Fun Math Problems/Brainteasers:

On average, how long do we need to roll a die and sum the rolls until the sum is prime? Moreover, what if we roll until the sum is composite?

(Putnam 2016): Let A be a 2n×2n matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability 1/2. Find the expected value of det(A − A^t) (as a function of n), where A^t is the transpose of A.

Find the number of ordered 64-tuples (x0, x1,…, x63) such that x0, x1,…, x63 are distinct elements of {1,2,…,2017} and x0 +x1 +2x2 +3x3 +···+63x63 is divisible by 2017.