I just join this forum and I really appreciate your work. This trick is awesome and I will try this one as soon as possible .
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I'm not sure what is meant by a ' general rule ', but if it's simply the maths behind this trick, that's pretty easy.
First note that if you know the bottom card of a pile, then you know how many cards there are in the pile, simply subtract from 11.
If for example the bottom card is a 4, then you count 4,5,6,7,8,9,10. That's 7 cards and equals 11 - 4.
Suppose then that the bottom cards of the three chosen piles are a, b and c.
The total number of cards in the three piles will be (11 - a) + (11 - b) + (11 - c) = 33 - (a + b + c), in which case the number of remaining cards will be 52 - {33 - (a + b + c)} = 19 + (a + b + c).
Subract 19 and you are left with (a + b + c), the sum of the bottom cards in the three piles.
I'm still not sure what it is that are looking for. It's easy enough to extend this to multiple packs, multiple stacks and a different top number (<=13), but it would seem that you are always going to arrive at some formula of this type.