Some math behind the Zener cards
The Zener cards are a deck of nm cards where each of n symbols is depicted on exactly m cards. The following experiment ($n=m=5$) has been used since the early ‘30s to test for extrasensory perceptions: the alleged telepath tries to guess cards one at a time, receiving some feedback after each attempt, until there are no cards left. The total number of correct guesses can be thus interpreted as a measure of their psychic powers. A very first step in the analysis of such experiments is to determine the optimal (non-psychic) expected score $S_{n,m}$. After an overview of the problem, I will focus on joint work with Steinerberger on the complete feedback case, where we determine the leading and next-to-leading order for $S_{n,m}$ in a wide range of regimes. In particular, this includes the case $n=m$, answersing a conjecture of Diaconis.