How many numbers must be selected from the set How many numbers must be selected from the set $\{1, 3, 5, 7, 9, 11, 13, 15 \}$ to guarantee that at least one pair of these numbers add up to $16$? Explain
your answer.
Just count the pairs that do add up to 16:

$$1+15 = 16, \quad3+13=16,\quad \dots$$ 

Then just ensure you choose enough numbers to guarantee that you have both numbers in one of these pairs.
solution pairs: (1,6) (2,5) (3,4)
    hole #1: for numbers 1 2 3
    hole #2: for numbers 4 5 6

111  1 
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#1   #2

If you pick 4 numbers you are guaranteed to hit a solution