There are $13$ white, $15$ black, $17$ red chips on a table. In one step, a person may choose $2$ chips of different colors and replace each one by a chip of the third color. Can all chips become the same color after some steps?
No, this is impossible. Note that the pairwise differences between the three colors are 2, 2, and 4. Furthermore, the operation described—turning $(a, b, c)$ into $(a+2, b-1, c-1)$—changes the differences between any two colors by either 3 or 0. In order to obtain all chips of a single color, we need counts $(45, 0, 0)$, which has a pair with difference 0. However, it is not possible to get from either 2 or 4 to 0 in steps of 3, so this cannot be done.