In Peter Winkler's Mathematical Puzzles A Connoisseur's Collection, he posed Fitch Cheney's card trick problem as follows.
His solution for the last question concerning the $124$-card, rather than the original $52$-card version is as follows.
However, I find myself puzzled by the solution. What is $k$? Is this a typo which is supposed to be $j$ and presumably $x=c_j$ the card Dorothy pulls out? We know neither $j$ or $k$ nor $c_k$ or $c_j$ or $x$. We need that to position $x$ in one of the modulo of $5$ before we can pick the exact position from amongst the $4!$ possibilities within each modulo class. Can someone explicate the solution?
Note: I understand the original $52$ card solution. Please do not explain that basic version.
Epilogue: I found Michael Kleber's The Best Card Trick. It gives a clear presentation. However, the new numbering system modulo 5 is best explained by the answer of @LonzaLeggiera below. This answer also has some good references on this problem.
Best Answer
Apart from the confusing description of what it is that "David needs to find", the trick won't work as described because of an error in the way Winkler indexes the cards chosen by the stranger. If you're going to use a sum mod $5$ as an index, then you need to index the cards as $\ c_0, c_1, \dots, c_4\ $ rather than $\ c_1, c_2, \dots, c_5\ $. If you use the latter indexing, which is what Winkler seems to assume later on in his explanation, then Dorothy has to choose the card $\ c_{j+1}\ $ rather than $\ c_j\ $, and David has to obtain a number $\ x\ $ such that $\ x\equiv -s + k \hspace{-0.3em}\mod 5\ $ and $\ x = c_{k+1}\ $, which is what I'll assume in the rest of this answer.
If $\ b_0<b_1<\dots\ <b_{119}\ $ are the cards remaining in the deck after the four which Dorothy hands to David have been removed from it, and $\ x= c_i\ (i=j+1)\ $ is the face value of the card Dorothy removes from the five chosen by the stranger before handing the remaining four to david, then $\ x=b_{x-i+1}=b_{x-j} $. Then if $\ s\ $ is the mod $5$ sum of the face values of the four cards Dorothy hands to David, $\ \sigma\equiv -s\equiv x-j\hspace{-0.3em}\mod 5\ $, and $\ d\ $ the unique number in the set $\ \left\{0,1,\dots,23\right\}\ $ such that $\ x-j=5d+\sigma\ $, Dorothy rearranges the cards in the order that she has prearranged with David to represent the number $\ d\ $. David can recover $\ d\ $ by inspecting the arrangement of the cards, and $\ \sigma\ $ by calculating the mod $5$ sum of their face values, so he can calculate the value of $\ x-j\ $ and $\ x=b_{x-j}\ $.