I must have seen this trick a couple of dozen times as a kid and I never found it even slightly impressive. It just seemed like there was a bit of sorting or memorization at play and I never felt the urge to figure out how it worked.
Fan out 21 cards for your mark and have him pick a card—often he will pick up the card and show it to a group while the magician is blindfolded. The magician then deals out the cards face-up into three piles and had the mark point to the pile containing the chosen card. The pile chosen is placed between the other two piles and then the cards are dealt into three piles again, etc. The cards are dealt 3 times and each time the pile chosen is placed into the middle.
The whole reason for this post is that I came upon this trick by accident while looking for other information on the web. The trick was not explained on the site except that it was described as above and posited that the chosen card would be the 10th one in the pile. For some reason, I decided to work out exactly how the trick works. I did this and found that it is not the 10th card, but rather the 11th which will be the chosen card! What is odd is that on the website, there were a few comments which purported to show mathematically that it is the 10th card. Maybe these folks are all in on the real joke, which is to make sport of some novice magician. Who knows? Maybe they are just idiots.
We shall number the cards when they are all stacked up as 1-21, with 1 on top and 21 being the bottom card. After the first deal and pick, the chosen card will be 8-14 (the 7 middle cards). When the cards are dealt again, the stacks will have 2, 2, or 3 cards each before the middle stack cards are put on top. So the middle cards will end up in positions 3, 4, or 5 in the three new stacks. When the cards are all re-stacked with the chosen pile in the middle the chosen card will be in positions 10, 11, or 12. When the cards are dealt for the last time, the first nine cards will form three stacks of three each, the chosen card will be the 4th card in whichever pile contains it. When I practiced the trick on my wife and kids, I just chose the 4th card from the pile they selected. If you were to stack the cards, it would be the 11th card since 7+4=11.
I thought about this trick and I think it would also work for up to 27 cards rather than the 21 used here, but I haven’t tried this # yet, but the chosen card would be card # 14 by my calculations.