If Girl #1 had the box labeled BBB, she would have known the remaining ball in that box was white. (The ball could not be black, otherwise the box would be correctly labeled.)
Also, if she had the box labeled BBW, she would have known the remaining ball in that box was black for the same reason.
If she had the other two labeled boxes, she could not have known the color of the remaining ball since it could have been either one.
However, since she did know the color of the remaining ball, she must have the box labeled either BBB or BBW. We don't yet know which is correct.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBB W, if BBB
BBW B, if BBW
 If Girl #2 had the box labeled BBW, she would have known the remaining ball in her box was white. (The remaining ball could not be black, otherwise the box would be correctly labeled.)
Also, if she had the box labeled BWW, she would have known the remaining ball in that box was black for the same reason.
If she had the other two labeled boxes, she could not have known the color of the remaining ball since it could have been either one.
And again, since she did know the color of her remaining ball, her box label must be either BBW or BWW. We don't yet know which is correct.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBB W, if BBB
BBW B, if BBW

2 BW BBW W, if BBW
BWW B, if BWW
 Girl #3 drew two white balls from her box and was not able to deduce the color of the remaining ball. This means she could not have had the box labeled WWW otherwise she would have known that the remaining ball could only be black.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBB W, if BBB
BBW B, if BBW

2 BW BBW W, if BBW
BWW B, if BWW

3 WW not WWW
 Girl #3 could also not have had the box labeled BWW since she then would have known her remaining ball could only have been white.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBB W, if BBB
BBW B, if BBW

2 BW BBW W, if BBW
BWW B, if BWW

3 WW not WWW
not BWW
 So, if Girl #3 could not have had the box labeled WWW or BWW, she must have had either the box labeled BBB or BBW.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBB W, if BBB
BBW B, if BBW

2 BW BBW W, if BBW
BWW B, if BWW

3 WW BBB
BBW
 So Girl #4 knows, since no one else can possibly have the box labeled WWW, then she must have it.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBB W, if BBB
BBW B, if BBW

2 BW BBW W, if BBW
BWW B, if BWW

3 WW BBB
BBW

4  WWW 
 Since Girl #1's box was not the box that contained the three white balls, nor was Girl #2's and since it can't be Girl #4's, (otherwise the label would be correct) then it can only be Girl #3's box that contains the three white balls.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBB W, if BBB
BBW B, if BBW

2 BW BBW W, if BBW
BWW B, if BWW

3 WW BBB W
BBW

4  WWW 
 Since Girl #2 is the only box that could possible have the label BWW, then she must have it.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBB W, if BBB
BBW B, if BBW

2 BW BWW B

3 WW BBB W
BBW

4  WWW 
 Since Girl #2's box contains the BlackBlackWhite ball combination, this combination can't be in Girl #1's box. Thus, Girl #1's box must have the BlackBlackBlack ball combination, and consequentially, the BBW label.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBW B

2 BW BWW B

3 WW BBB W
BBW

4  WWW 
 Since we know Girl #1 now has the Box Label of BBW, Girl #3 cannot have it.girl balls Incorrect Ball
# drawn box label remaining

1 BB BBW B

2 BW BWW B

3 WW BBB W

4  WWW 
 Girl #4's box obviously then contains two white balls and one black ball since no other box has that combination in it.
