Suppose the values of the cards at the bottom of the three piles are

a

b

and

c

The three piles contain $14 - a$ , $14 - b$ and $14 - c$ cards respectively, so the magic counting pile contains $52 - (14 - a) - (14 - b) - (14 - c) = 10 + a + b + c$ cards.

Suppose we turn over piles a and b. We count out $a + b + 10$ cards, leaving exactly $c$ in the magic counting pile.

Ta-da! Delightful.