To solve the problem, we need to analyze the game played by players A and B. Here’s a step-by-step solution:
### Step 1: Understand the Game Rules
- Player A starts and can only call out odd numbers.
- Player B can only call out even numbers.
- Each player must call out a number greater than the previous number called (except for A's first turn).
- The game ends when a player cannot call out a valid number.
### Step 2: Identify the Range of Numbers
- Both players can only call out numbers from 1 to n.
- A can call odd numbers: 1, 3, 5, ..., up to the largest odd number ≤ n.
- B can call even numbers: 2, 4, 6, ..., up to the largest even number ≤ n.
### Step 3: Determine the Possible Moves
- If n is odd, A can call odd numbers up to n, and B can call even numbers up to n-1.
- If n is even, A can call odd numbers up to n-1, and B can call even numbers up to n.
### Step 4: Count the Possible Games
Let’s denote the number of possible games for a given n as P(n). We can derive P(n) based on previous values:
1. **Base Cases:**
- P(1) = 1 (A calls 1, B cannot call anything)
- P(2) = 1 (A calls 1, B calls 2)
- P(3) = 2 (A can call 1 or 3, B can only call 2)
- P(4) = 3 (A can call 1, 3; B can call 2, 4)
- P(5) = 5 (A can call 1, 3, 5; B can call 2, 4)
- P(6) = 8 (A can call 1, 3, 5; B can call 2, 4, 6)
2. **Recursive Relation:**
- For n ≥ 2, we can express P(n) as:
\[
P(n) = P(n-1) + P(n-2)
\]
This is because:
- If A calls an odd number (1, 3, ..., n-1), then B has the remaining options.
- If A calls the largest odd number (n), then B has no options left.
### Step 5: Calculate P(n) for Specific Values
Using the recursive relation, we can calculate P(n) for various values of n:
- P(7) = P(6) + P(5) = 8 + 5 = 13
- P(8) = P(7) + P(6) = 13 + 8 = 21
- P(9) = P(8) + P(7) = 21 + 13 = 34
- P(10) = P(9) + P(8) = 34 + 21 = 55
### Step 6: Verify the Options
Now we can check which options are correct based on the calculated values:
- For n = 6, P(6) = 8 (Correct)
- For n = 8, P(8) = 21 (Correct)
- For n = 10, P(10) = 55 (Correct)
- For n = 2, P(2) = 1 (Incorrect if stated otherwise)
### Conclusion
The correct options based on the values calculated are:
- A (n = 6, P(6) = 8)
- C (n = 8, P(8) = 21)
- D (n = 10, P(10) = 55)
