Great, well done!

Slight niggle: I disagree “there is no slight of hand”. You have to order the piles differently without people noticing. There is no trickery in the sense that you aren’t hiding the chosen card or forcing a choice.

When we did this at Nottingham MathsJam I also showed the 21 cards trick which is similar except you always put it in the middle with 10 cards either side. I have a note that we thought the trick would work in base $b$ for $x<b^n$ cards for odd $x$ if you always wanted to put it in the middle.

Thanks.

The 21-card version leads much more quickly to an understanding of what’s going on. Actually, I think the condition is that if you are dealing out $p=2m+3$ piles, $m\ge 0$, then decks of $(2n+1)p$ cards, $n\ge 0$, will work in $n+1$ goes. A proof by induction should work.

Actually, I’d better clear this up to make sure I’ve understood: your 21-card version doesn’t ask the mark for a number, and just puts their card in the middle of the deck after a certain number of goes?

Sorry, $n+1$ goes is way too generous. Something like $\lceil {\log }_p(2n+1)p\rceil$ goes.

Actually, I’d better clear this up to make sure I’ve understood: your 21-card version doesn’t ask the mark for a number, and just puts their card in the middle of the deck after a certain number of goes?

Correct. I was shown this at the same time. Actually, I think it was through copying the 21 version that I worked out how to repeat the 27 version.