  We prove the Murphy and Cohen's conjecture that the maximum number of
collisions of n + 1 elastic particles moving freely on a line is n(n+1)/2 if no
interior particle has mass less than the arithmetic mean of the masses of its
immediate neighbors. In fact, we prove the stronger result that, for the same
conclusion, the condition no interior particle has mass less than the geometric
mean, rather than the arithmetic mean, of the masses of its immediate neighbors
suffices.
