How many numbers must be selected from the set {1, 2, 3, 4, 5, 6} to guarantee that at least one pair of these numbers add up to 7?

Step 1 (Identify pairs that sum to 7): Within the set $\{1,2,3,4,5,6\}$, we partition the numbers into pairs (pigeonholes) that sum to exactly 7:

1. $\{1,6\}$

2. $\{2,5\}$

3. $\{3,4\}$

There are exactly 3 such pairs.

Step 2 (Apply Pigeonhole Principle): If we select numbers from the set, we want to guarantee that we choose both numbers from at least one of these 3 pairs.

• If we select 3 numbers, we could select at most 1 number from each of the 3 pairs (e.g., $\{1,2,3\}$), which does not guarantee a pair summing to 7.

• If we select 4 numbers, by the Pigeonhole Principle, since we are distributing 4 selections into 3 pairs (pigeonholes), at least one pair must contain 2 selections. This guarantees that at least one pair adds up to 7.

Step 3 (Conclusion): The minimum number of selections required is exactly 4.