We study two problems related to top to random shuffling. First, we show that for an O(n ln( n)) top to random shuffled deck, there exists a best no feedback card guessing strategy, i.e., a strategy maximizing the expected number of correct guesses without revealing information to the guesser. Moreover, using the best no feedback guessing strategy, the difference between the expected number of correct guesses for a deck that is O(n ln(n)) top to random shuffled and a deck that is uniformly distributed can be made arbitrarily small.

Second, we study the expected value and the variance of the number of fixed points of a permutation that is obtained after an arbitrary number of top to random shuffles and provide closed formulas for both. Two different methods of proofs are provided, one combinatorial and another one using the distribution of the eigenvalues among the irreducible representations of S_n. Since the number of fixed points is a feature of a permutation, it is natural to expect that with O( n ln(n)) shuffles it will converge to Poisson(1). However, we show that O(cn) top to random shuffles (contrary to riffle shuffles) are not enough to achieve convergence to a Poisson(1).