Dom and Mina really like playing dominoes. They have some unusual sets of dominoes, including one particularly rare set comprising dominoes with numbers, ranging from 1 to 100, rather than spots. There is exactly one domino of each possible configuration, that is, for every pair of distinct whole numbers \(m\) and \(n\) between 1 and 100 there is a domino with \(m\) on one side and \(n\) on the other. There are no dominoes with the same number on each side. For example, the domino 2-5 is the same as 5-2, and there is just one of these. There are no blanks.

Dom and Mina arrange the dominoes into the longest possible single line of dominoes (as is usual with dominoes, the top of one domino is allowed to touch the bottom of another domino only if the numbers on the top of the first domino and the bottom of the second domino are the same). How many dominoes are left over?

MathsBombe Competition 2019 is organised by the The School of Mathematics at the
University of Manchester.

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