Numbers are drawn from 973 integers, 27 to 999, at random. At least how many numbers need to be drawn to ensure that there are two numbers drawn whose sum is 659?

We need to determine the minimum number of integers that must be drawn from the set of integers between 27 and 999 such that we are guaranteed to have two numbers whose sum equals 659.

### Step 1: Define the set of integers
The numbers are drawn from the set of integers \( S = \{27, 28, 29, \dots, 999\} \). This is a total of 973 integers.

### Step 2: Consider pairs of numbers
For any two numbers \( x \) and \( y \) from this set, we are looking for pairs such that their sum is 659:
\[
x + y = 659
\]
Thus, \( y = 659 - x \).

### Step 3: Identify valid pairs
We need both \( x \) and \( y \) to be in the range of the integers between 27 and 999. Specifically:
- If \( x = 27 \), then \( y = 659 - 27 = 632 \).
- If \( x = 28 \), then \( y = 659 - 28 = 631 \).
- If \( x = 29 \), then \( y = 659 - 29 = 630 \).
- And so on, continuing this pattern.

The smallest valid \( x \) is 27 and the largest valid \( x \) is \( 659 - 27 = 632 \). So, we are only concerned with pairs where \( x \in \{27, 28, \dots, 632\} \) and \( y = 659 - x \).

### Step 4: Count the number of pairs
Each number \( x \in \{27, 28, \dots, 632\} \) pairs uniquely with \( y = 659 - x \), which is also in the set \( S \). The numbers from 27 to 632 form 606 pairs, since:
\[
632 - 27 + 1 = 606
\]
Thus, there are 606 valid pairs of numbers whose sum is 659.

### Step 5: Apply the pigeonhole principle
The pigeonhole principle tells us that if we have more items than containers, then at least one container must hold more than one item. In this case, the "items" are the numbers we draw, and the "containers" are the pairs whose sum is 659.

If we draw 607 numbers, we are guaranteed to have two numbers from the same pair because there are only 606 distinct pairs whose sum is 659. Therefore, the minimum number of numbers we must draw to ensure that two of them sum to 659 is:

\[
\boxed{607}
\]

### Would you like more details or have any questions?

Here are five related questions for further practice:
1. What if the sum required was 700 instead of 659?
2. How many numbers need to be drawn to ensure three numbers sum to 659?
3. What if the integers were drawn from a smaller range, say 50 to 500?
4. How would the answer change if the

Explore how to determine the minimum number of numbers that need to be drawn from 973 integers (ranging from 27 to 999) to ensure that two numbers sum to 659. This solution uses number theory and the pigeonhole principle to provide the answer of 607.

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