Question 15
Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not both appear for any $i$ and $j.$ Let $D_{40}$ be the set of all dominos whose coordinates are no larger than $40.$ Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$