17.6. Question 9

Try these first

1. 1.

   List all distinct pairs of integers from the set {1, 2, 3, 4, 5, 6, 7, 8} that sum to 9.

   Hint

   Pair numbers that add up to 9.

2. 2.

   If you select the integers 1, 2, 4, 5, what pair (if any) will sum up to 9?

   Hint

   Consider the remaining integers in your set.

3. 3.

   How many distinct pairs can sum to 9 from the set {1, 2, 3, 4, 5, 6, 7, 8}?

   • 2
   • 4
   • 8

   Hint

   Count available combinations.

4. 4.

   If you choose the integers 1, 4, and 7, will there be at least one pair summing to 9?

   • True
   • False

   Hint

   Think about missing integers to round out your selected numbers.

5. 5.

   If you randomly select 7 integers from the set {1, 2, 3, 4, 5, 6, 7, 8}, can you guarantee at least one sum of 9? Justify your answer using the pigeonhole principle.

   Hint

   Refer back to the pairs you have established.

6. 6.

   If a new integer 9 was added to the set, how would that change the original proof regarding sums to 9? Discuss and reason.

   Hint

   Think about how to create new pairs with the existing numbers.