How does this card trick work?

Question

Pick a card from the deck and keep it secret. Double the face value of the card (aces = 1, jacks = 11, queens = 12, and kings = 13). Add 3 to the result. Multiply this by 5. Add 1 if the card is a club, 2 if it is a diamond, 3 if it is a heart, and 4 if it is a spade. Ask them to tell you their number.

To predict the card, subtract 15 from the final total. The right digit of the answer represents the suit of the card (1 = club, 2 = diamond, 3 = heart, 4 = spade). The left digit or digits is the number value of the card. For example, if their result is 83, the card is the 8 of hearts. If the result is 134, the card is the king of spades. Can you figure out how this trick works?

Thank you, Liz

Answer 1

Call the face value $f$ and the suit value $s$.

• You start with $f$
• Double it to $2f$
• Add three to give $2f+3$
• Multiply by five to give $10f+15$
• Add the suit value to give $10f+s+15$
• Subtract fifteen to give $10f+s$

Since $1 \le s \le 4$ the right hand digit is $s$ and the tens' digit (and perhaps hundreds' digit) show $f$.