A bag contains 7 red discs, 8 blue discs and 9 yellow discs. Two discs are drawn at random from the bag. If the discs are the same colour then they are put back into the bag. However, if the discs are different colours then they are removed from the bag and a disc of the third colour is placed in the bag. This procedure is repeated until there is only one disc left in the bag or the only remaining discs in the bag have the same colour.
What colour is the last disc (or discs) left in the bag?
Denote the number of red disks by R, blue disks by B and yellow disks by Y.
We will prove that at any point of the game either:
(a) R, Y are both odd and B is even or
(b) R, Y are both even and B isodd.
At the start, the game is in the state (a) above. If we draw two discs of the same colour, the total number of disks of each colour does not change. If we draw two discs of different colours, each of R, Y, B changes by one, changing the parity of all three of these values, hence moving the game between states (a) and (b).
The game cannot end in state (a) as it means R, Y >0, so it must end in state (b). As in state (b), B cannot be zero, the last disc (or discs) left in the bag is blue.
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