Twenty pairs of integers are formed using each of the integers $1,2,3, \ldots, 40$ once. The positive difference between the integers in each pair is 1 or 3. (For example, 5 can be paired with 2, 4, 6 or 8.) If the resulting differences are added together, the greatest possible sum is
(A) 50
(B) 54
(C) 56
(D) 58
(E) 60

## Part C: Each correct answer is worth 8.
Twenty pairs of integers are formed using each of the integers $1,2,3, \ldots, 40$ once. The positive difference between the integers in each pair is 1 or 3 . (For example, 5 can be paired with 2, 4, 6 or 8.) If the resulting differences are added together, the greatest possible sum is
(A) 50
(B) 54
(C) 56
(D) 58
(E) 60

## Solution

Since we have twenty pairings, it is possible to have nineteen differences of 3 and one difference of 1 . The maximum sum of these differences is thus, $3(19)+1=58$. The pairings can be achieved in the following way: $\{(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22)$, $(20,23),(21,24),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(37,40),(38,39)\}$. Note that there is just one pair, $(38,39)$, that differs by one.

ANSWER: (D)

## Part C: Each correct answer is worth 8.