Two-Letter Card Game

Two-Letter Card Game

What is this problem about?

"Two-Letter Card Game" is an analytical counting problem. You are given a set of cards, where each card has two letters. You need to determine how many pairs of cards can be formed such that the two cards in the pair share exactly one letter in the same position. For example, "ab" and "ac" share the first letter 'a' but have different second letters. This problem requires efficient counting and grouping of strings.

Why is this asked in interviews?

Amazon often uses this "Two-Letter Card Game interview question" to test a candidate's ability to use "Hash Tables" for multi-criteria counting. It involves "Enumeration" and combinatorics. Interviewers want to see if you can calculate the number of valid pairs without a $O(N^2)$ comparison, which would be too slow if you have thousands of cards. It's a test of your ability to handle "Combinatorial Explosions" using pre-computed frequencies.

Algorithmic pattern used

The "Array, Hash Table, Counting, Enumeration, String interview pattern" is the way to go.

1. Store the frequency of every two-letter combination in a map.
2. For each unique card XY:
   • Find all cards that look like _Y (same second letter, different first).
   • Find all cards that look like X_ (same first letter, different second).
   • Multiply the current card's frequency by the frequencies of these "near-matches" to get the pair count.
3. Be careful to divide the final sum by 2, as each pair is counted twice.

Example explanation

Cards: ["ab", "ac", "ab", "bc"]

• Frequency: {ab: 2, ac: 1, bc: 1}
• For "ab" (count 2):
  • Matches "a_": "ac" (count 1). Pairs = $2 \times 1 = 2$.
  • Matches "_b": None (different from "ab").
• For "ac" (count 1):
  • Matches "a_": "ab" (count 2). Pairs = $1 \times 2 = 2$.
  • Matches "_c": "bc" (count 1). Pairs = $1 \times 1 = 1$. Total = $2 + 2 + 1 = 5$ (counted twice). Sum = 10. Result = 5.

Common mistakes candidates make

A common mistake is forgetting that pairs must share exactly one letter. This means you must exclude cases where both letters match (i.e., the same card type). Another error is over-counting pairs or not handling the frequency of identical cards correctly. Using a simple $O(N^2)$ approach will also fail for large datasets.

Interview preparation tip

To solve the "Two-Letter Card Game coding problem," practice "Frequency-Based Counting." Instead of iterating over every item, iterate over the "Possibility Space" (which for two letters is just $26 \times 26$). This shift in perspective is key to optimizing many string and counting problems.