A bag contains 3 red, 4 white, and 5 blue balls. Two balls of different
color are removed from the bag and replaced with two balls of the third color.

Prove that no matter how many times this procedure is repeated, it is impossible for all twelve balls in the bag to be the same color.
  

(In reply to Yet another approach. by Jer)

Like Jer, I've applied the "reverse engineering" approach, trying to define what common feature is shared by the "reachable" triplets as opposed to "unreachable" ones.

Helas - I did not consider "going mod 3."

Comparing the 3 approaches presented so far I feel that to say  that Paul's solution is much more elegant is an understatement. 

It is the only general solution , valid for 3, 4, 5 triplet as well as for 333, 344,  338. (Just imagine finding all possible paths for the new triplet!) 

Excellent proof, Paul!

Nice problem (I've rated it 4), Bractals!

Posted by Ady TZIDON on 2015-07-30 01:59:48