TL;DR version: With the magician's back turned, a spectator shuffles a standard deck of $n=52$ playing cards and selects and arranges $m=27$ of them face-up in a row left to right. The magician's assistant turns one of these cards face-down, leaving the ordered arrangement otherwise undisturbed. The magician turns around, observes this arrangement with $k=m-1=26$ face-up cards, and announces the single face-down card. How do the magician and assistant perform this trick?
Details: I'm trying to generalize the ideas behind Fitch Cheney's "5-card trick," where a spectator selects $m=5$ cards, the assistant arranges $k=4$ of them in a row left to right to present to the magician, who announces the remaining card. This trick is feasible if and only if ${n \choose m} \leq (n)_k$ (where the RHS is the falling factorial); that is, considering the set $U$ of possible $m$-card selections by the spectator and the set $V$ of possible $k$-card arrangements that may be presented to the magician, the obvious necessary condition that $|U|\leq|V|$ is also sufficient (this follows from the bi-regularity of the corresponding bipartite graph with vertex sets $U$ and $V$).
There is a very simple "calculate-in-your-head" method for performing that original 5-card trick. But the above inequality suggests that the trick should be feasible in principle with a larger deck, using a maximum of $n=124$ cards; discussions suggest that Berlekamp and Kleber have actually performed this variant.
The proposed "27-card" variant above is interesting in that the spectator gets to arrange the selected cards; all the assistant can do is determine which of the cards to turn over. A similar counting argument shows that this trick should be feasible in principle, since $(n)_m \leq (n)_k{m \choose k}$: the LHS counts ways for the spectator to select and arrange $m=27$ cards from the deck of $n$, and the RHS counts configurations that the magician may observe with $k=m-1=26$ face-up cards.
But I'm having difficulty coming up with an actual explicit construction of an appropriate mapping, even a formula that might be suitable for a computer version of the trick, let alone mental calculation.
(For another example demonstrating the idea: a spectator rolls $m=6$ dice each with $n=6$ sides and arranges them in any desired order left to right, the assistant covers some of the dice, leaving the $k<m$ other dice showing, and the magician announces the value on the covered di(c)e. In this case, the trick is feasible if and only if $n^m \leq n^k{m \choose k}$. For example, $(n,m,k)=(10,10,9)$ corresponds to writing down a 10-digit number, covering one digit, and announcing that covered digit. This can be done by indexing the digit positions 0 through 9: the assistant covers the digit in position $s$, where $s$ is the sum of the digits modulo 10. The magician computes the covered value as $s-t \mod 10$, where $t$ is the sum of the visible digits.)
