Question

Three boxes each contain a number of billiard balls. One box contains only even-numbered balls, one box contains only odd numbered billiard balls, and the third box contains a mixture of odd and even numbered balls. All the boxes are mislabeled. By
selecting one ball from only one of the boxes, can you correctly label the three boxes? Explain why or why not.

Answer 1

If all 3 boxes are mislabeled, then selecting a ball from box incorrectly labelled as mixture of odd and even, 2 cases arise
1.if the ball is odd and the box is incorrectly labelled as mixture of odd/even then the box has to be the box of odd numbered balls
2.if it is even, the box is in fact the box of even numbered ball
See if you can get anywhere from there

Answer 2

continuing from case 1,
if the box incorrectly labelled as mixture of odd/even is actually box of odd numbered ball,
then the box incorrectly labelled as even numbered balls must be either box of mixture of odd/even or it must be the box of even.But it cannot be the box of even because all boxes are mislabelled, thus it must be the box of mixture of odd/even and thus the box incorrectly labelled as odd is infact the box of even numbered balls
Case 2
Since the box incorrectly labelled as mixture of odd/even is actually box of even numbered balls, then the box incorrectly labelled as odd numbered balls must be either box of mixture of odd/even or it must be box of odd.It can't be box of odd as labels are not correct, thus the box marked as odd is actually the box of mixture of odd/even balls and finally the box marked as even is actually the box of odd balls.

Answer 3

Thus you an correctly label if you choose a ball from the box incorrectly marked as the mixture of odd/even