Answer to Question #92695 in Combinatorics | Number Theory for Felomina

Question

Suppose you decide to represent the cards in a standard deck using the numbers 1 through 52. So for this, the cards are arranged so that the clubs come first, then the diamonds, then the hearts and then the spades. Also within a suit, the order of the cards is ace, then the cards 2 through 10, then the jack, then the queen, then the king. So after arranging the cards in this manner, the ith card is assigned the number i. Thus the king of spades will get the number 52, and the ace of clubs the number 1.

1.What number does the 9 of spades get?

2. Suppose some card has number n. I would like to know if it is a diamond. Which of the following tests will tell me if this is the case?
Options

1.n > 26

2.ceiling(n/4) = 2

3. n mod 13 = 0

4ceiling(n/13) = 2

Answer 1

1) Cards whose numbers are less than number of the nine of spades are:

all clubs, all diamonds, all hearts, ace of spades, two of spades,...,eight of spades,

total 13 + 13 + 13 + 8 = 47 cards. So nine of spades gets number 47 + 1 = 48.

2) Ace of diamonds gets number 13 + 1 = 14, there are 13 cards with smaller numbers (all clubs),

king of diamonds gets number 13 + 13 = 26.

If the card is diamond then its number $14\leq$ n$\leq 26$, therefor 1<14/13 $\leq$ n/13$\leq 26/13=2$,

hence ceiling(n/13) = 2.

Answer: 1) 48, 2) ceiling(n/13) = 2.