How many odd integers are greater than the integer x and less than the integer y? There are 12 even...

## Understanding the Question

We need to find the exact count of odd integers in the open interval (x, y), where x and y are integers. In other words, how many odd numbers are strictly between x and y (not including x or y themselves)?

For this question to have a sufficient answer, we need either:
- The exact count of odd integers, or
- Enough information to calculate this count uniquely

### Key Insight

Here's the crucial insight: consecutive integers alternate between odd and even. This means if we know the total number of integers in an interval, we can determine exactly how many are odd and how many are even. The distribution depends on whether we start with an odd or even number, but for any given count of consecutive integers, there are only one or two possible distributions.

## Analyzing Statement 1

**Statement 1 tells us**: There are 12 even integers between x and y.

This gives us a piece of the puzzle, but we're missing crucial information. We know there are 12 even integers, but we don't know:
- The total number of integers between x and y
- The specific values of x and y
- Whether the sequence starts with odd or even

Let's test different scenarios to see if we can get different counts of odd integers:

**Scenario 1**: If x = 0 and y = 25
- Integers between them: 1, 2, 3, ..., 24
- Even integers: 2, 4, 6, ..., 24 → That's 12 even integers ✓
- Odd integers: 1, 3, 5, ..., 23 → That's 12 odd integers

**Scenario 2**: If x = 0 and y = 26
- Integers between them: 1, 2, 3, ..., 25
- Even integers: 2, 4, 6, ..., 24 → That's 12 even integers ✓
- Odd integers: 1, 3, 5, ..., 25 → That's 13 odd integers

Different scenarios give us different counts of odd integers (12 vs 13), even though both have exactly 12 even integers.

Statement 1 is **NOT sufficient**.

**This eliminates choices A and D.**

## Analyzing Statement 2

**Now let's forget Statement 1 completely and analyze Statement 2 independently.**

**Statement 2 tells us**: There are 24 integers between x and y.

This is powerful information. We have exactly 24 consecutive integers. Let's think about what happens with any set of 24 consecutive integers.

Since consecutive integers alternate between odd and even, let's consider the two possible cases:

**Case 1**: If the first integer after x is odd
- Pattern: odd, even, odd, even, odd, even... (24 integers total)
- Count: 12 odd, 12 even

**Case 2**: If the first integer after x is even
- Pattern: even, odd, even, odd, even, odd... (24 integers total)
- Count: 12 even, 12 odd

Notice something beautiful? In both cases, with 24 consecutive integers (an even count), we get exactly 12 of each type. This is because consecutive integers alternate perfectly, and \(24 \div 2 = 12\).

No matter where we start—whether x is odd or even, positive or negative—any interval containing exactly 24 consecutive integers will always have **exactly 12 odd integers and 12 even integers**.

**[STOP - Sufficient!]** We can determine the exact count: 12 odd integers.

Statement 2 is **sufficient**.

**This eliminates choices A, C, and E.**

## The Answer: B

Statement 2 alone tells us there are exactly 12 odd integers between x and y, while Statement 1 alone cannot determine this count.

**Answer Choice B**: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."