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

Modern Puzzles by Henry Ernest Dudeney: $198$

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.

1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Problems Concerning Games: $198$. -- A Card Trick
1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Miscellaneous Puzzles: $183$. A Card Trick