Guaranteeing a Sum of 16 with Pairs from 1 to 15: A Pigeonhole Principle Approach

In this article, we delve into the mathematical problem of ensuring that a pair of numbers, selected from the set containing the numbers 1 to 15, will add up to 16. We will explore the solution methodically by employing the Pigeonhole Principle to reach a concrete outcome.

Identifying the Pairs

To solve this problem, we start by identifying all the pairs of numbers within the set {1, 2, 3, ..., 15} that sum to 16. We need to enumerate these pairs carefully:

(1, 15) (2, 14) (3, 13) (4, 12) (5, 11) (6, 10) (7, 9)

Note that 8 is not included in any pair since repeating the number 8 would not contribute to our goal of distinct pairs.

Counting the Pairs

From the list above, we can clearly see that there are 7 distinct pairs that sum to 16:

(1, 15) (2, 14) (3, 13) (4, 12) (5, 11) (6, 10) (7, 9)

Applying the Pigeonhole Principle

The Pigeonhole Principle is a fundamental concept in combinatorics that states that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. In our context, we can view each of these 7 pairs as a "pigeonhole."

If we select 8 numbers from the set {1, 2, 3, ..., 15}, we will necessarily pick at least one number from at least one of these 7 pairs. This ensures that at least one pair will add up to 16.

Conclusion

To guarantee that there is at least one pair of numbers that adds up to 16, we need to pick 8 numbers from the set containing the numbers 1 to 15. This is because 8 is one more than the number of distinct pairs (7) that sum to 16.

Mathematical Verification

To further validate this solution, we can consider a sample of the possible 2-digit pairs and their sums:

m . 16 105  # The number of pairs that sum to 167    # There are 7 different digit pairs that sum up to 16n . (1 2 3 4 5 6 7 8 9 10 11 12 13 14 15) {~ i. 15 2{. (m / n) # List the pairs that sum to 16│1 15│2 14│3 13│4 12│5 11│6 10│7 9│

From this verification, we can further conclude that any selection of 8 numbers from the set {1, 2, 3, ..., 15} will ensure that at least one of the pairs adds up to 16.