Henry Ernest Dudeney/Modern Puzzles/198 - A Card Trick/Solution

by : $198$

 * A Card Trick
 * Take an ordinary pack of playing-cards and regard all the court cards as tens.
 * Now, look at the top card -- say it is a seven -- place it on the table face downwards and play more cards on top of it, counting up to twelve.
 * Thus, the bottom card being seven, the next will be eight, the next nine, and so on, making six cards in that pile.
 * Then look again at the top card of pack -- say it is a queen -- then count $10$, $11$, $12$ (three cards in all) and complete the second pile.
 * Continue this, always counting up to twelve, and if at last you have not put sufficient cards to complete a pile, put these apart.
 * Now, if I am told how many piles have been made and how many unused cards remain over,
 * I can at once tell you the sum of all the bottom cards in the piles.
 * I simply multiply by $13$ the number of piles less $4$, and add the number of cards left over.
 * Thus, if there were $6$ piles and $4$ cards over, then $13$ times $2$ (i.e. $6$ less $4$) added to $5$ equals $31$, the sum of the bottom cards.
 * Why is this?
 * This is the question.

Solution
Let there be $n$ piles.

Let $p_k$ denote the number of cards in pile $k$.

Let $b_k$ denote the value of the bottom card.

Let $m$ be the number of cards left over.

We have that:
 * $p_k = 13 - b_k$

for all $1 \le k \le n$.

That is, the sum of all the bottom cards in the piles equals the number of piles minus $4$, all times $13$, plus the ones left over.