We have n guinea pigs placed on the vertices of a regular polygon with n sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, such direction is either "clockwise" or "anti-clockwise", and a velocity between 1km/h, 2km/h,..., and nkm/h, each one with a distinct velocity, and each guinea pig has a counter starting from 0. They start moving along the circumference with the assigned direction and velocity, everyone at the same time, when 2 or more guinea pigs meet a point, all of the guinea pigs at that point follow the same direction of the fastest guinea pig and they keep moving (with the same velocity as before); each time 2 guinea pigs meet for the first time in the same point, the fastest guinea pig adds 1 to its counter. Prove that, at some moment, for each 1≤i≤n we have that the i−th guinea pig has i−1 in its counter. combinatoricscombinatorial geometry