Assign a binary code in some orderly manner to the 52 playing cards. Use the minimum number of bits.
The correct answer and explanation is :
To assign a binary code to the 52 playing cards using the minimum number of bits, we need to calculate the number of bits required to uniquely represent each of the 52 cards.
Step 1: Determine the number of bits needed
The first thing we need to determine is how many unique combinations we need to represent. Since there are 52 distinct playing cards in a standard deck, the minimum number of bits required is the smallest integer ( n ) such that:
[
2^n \geq 52
]
Calculating powers of 2:
- ( 2^5 = 32 ) (not enough)
- ( 2^6 = 64 ) (sufficient)
Thus, 6 bits are required to represent 52 unique values. With 6 bits, we can represent 64 unique values, which is more than sufficient to cover the 52 cards.
Step 2: Assign binary codes to the 52 cards
A standard deck of 52 playing cards consists of 13 ranks (Ace, 2, 3, …, 10, Jack, Queen, King) and 4 suits (Hearts, Diamonds, Clubs, Spades). One way to assign binary codes to the cards is to use the following method:
- Rank encoding: The ranks can be numbered from 0 to 12 (where Ace = 0, 2 = 1, …, King = 12). This requires 4 bits since ( 2^4 = 16 ), and we only need to represent 13 ranks.
- Ace = 0000, 2 = 0001, 3 = 0010, …, King = 1100.
- Suit encoding: The suits can be numbered from 0 to 3 (Hearts = 0, Diamonds = 1, Clubs = 2, Spades = 3). This requires 2 bits since ( 2^2 = 4 ), which is enough to represent the 4 suits.
- Hearts = 00, Diamonds = 01, Clubs = 10, Spades = 11.
Step 3: Combine rank and suit binary codes
To uniquely represent each card, we combine the 4-bit rank code and the 2-bit suit code into a 6-bit binary code. For example:
- Ace of Hearts = 0000 00 (Rank 0, Suit 0)
- 2 of Diamonds = 0001 01 (Rank 1, Suit 1)
- King of Spades = 1100 11 (Rank 12, Suit 3)
Thus, each of the 52 cards can be uniquely identified with a 6-bit binary code, ensuring that the number of bits is minimized.
Conclusion
By using a 6-bit binary code, we can efficiently represent all 52 playing cards in a deck with the minimum number of bits. Each card is uniquely identifiable, and this approach guarantees the most efficient use of binary encoding for a standard deck of cards.