Well done Sara and Terence for cracking this
Cunning Card Trick.
"My solution starts from when you have the three piles plus the
Of the three piles, take the three bottom cards to be $x$, $y$ and
In the $x$ pile there will be $13 - x + 1$ cards.
This is because whatever the value of $x$, you counted on to $13$,
(if $x = 7$ you count 8, 9, 10, 11, 12, 13 (6 cards)) but you also
have the $x$ card which is why you add 1 more.
So an $x$ value of $7$ gives you $7$ cards in the pile, $8$ gives
you $6$, queen gives you $2$, etc.
So pile $x$ contains $14 - x$ cards, pile $y$ contains $14 - y$
cards and pile $z$ gives you $14 - z$ cards."
Using the fact that there are 52 cards
in a pack, Terence finished off the argument as follows:
"So, we have in total $42 - (x + y + z)$ cards in the piles, and so
$10 + x + y + z$ cards in the magic counting pile.
Suppose we deal y and z cards out, there will be just $10 + x$
cards left in the magic counting pile.
Dealing a further 10 leaves $x$ cards, which is the value of the
bottom card of the secret pile!
Alexander and James from The Batt Primary
School, Witney have just sent us this solution
Many thanks for such a clear