You have 3 cans. On is labelled “black” and contains only, one is labelled “white” and contains only white marbles, the last is labelled “mixed” and contains both black and white marbles.
All labels switched
Unfortunately, all the labels get switched so that none of the cans have their original label. How many marbles do you have to draw to determine which can is which?
You only need to draw one marble.
Proof
Draw a marble from the can labelled “mixed”.
If you draw a black marble it must be the “black” can since it cannot actually be the “mixed” can. Thus the can currently labelled “black” is the white can, since it cannot be either the “black” or “mixed” cans. Finally the can labelled “white” must be the mixed can.
Similarly if a white marble is drawn, the can labelled “mixed” is really “white”, the can labelled “white” is really “black” and the can labelled “black” is really “mixed”.
Two labels switched
What if only two of the labels were switched and one remained the same? How many marbles do you need to draw now?
You need to draw two marbles.
Proof
First note that one marble is not sufficient since there are 3 possible swaps. Single marble can only distinguish between two possibilities.
Start by drawing a marble from the can labelled “mixed”.
If a white marble draw a marble from the can labelled “black”. If this second marble is white, then the “mixed” can is correct, and the “black” label was swapped with the “white” label. If the second marble is black then the “black” can is correct and the “mixed” was swapped with the “white”.
If the first marble drawn was black then follow the above analysis with black and white reversed.