WakeUp wrote: Assume we have $95$ boxes and $19$ balls distributed in these boxes in an arbitrary manner. We take $6$ new balls at a time and place them in $6$ of the boxes, one ball in each of the six. Can we, by repeating this process a suitable number of times, achieve a situation in which each of the $95$ boxes contains an equal number of balls? Since $6\times 16 = 96$, we can put $16$ times $6$ balls in the boxes so that the number of balls in one of the boxes increases by two, while in all other boxes it increases by one. Repeating this procedure, we can either diminish the difference between the number of balls in the box which has most balls and the number of balls in the box with the least number of balls, or diminish the number of boxes having the least number of balls, until all boxes have the same number of balls.