# Dice and Division

Player $$A$$ and player $$B$$, are playing a game. They each have an unbiased 10-sided dice with faces labelled with the numbers $$1,2,3,...,10$$. To play, $$A$$ and $$B$$ simultaneously roll their dice. The number that $$A$$ rolls is denoted as $$m$$ and the number that $$B$$ rolls is denoted as $$n$$. If $$m = n$$ then they roll the dice again (ie. it is like the last roll didn't happen). If $$|m-n|$$ divides $$m+n$$ then $$B$$ wins, else $$A$$ wins. The probability that $$B$$ wins can be expressed as $$\dfrac{p}{q}$$ where $$p$$ and $$q$$ are coprime positive integers. What is $$p+q$$?

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