Boxes and Balls

Four girls were blindfolded and were each given an identical box, containing different colored balls.

One box contained 3 black balls. One box contained 2 black balls and 1 white ball. One box contained 1 black ball and 2 white balls. One box contained 3 white balls.

Each box had a label on it reading 'BBB' (Three Black) or 'BBW' (Two Black, One White) or 'BWW' (One Black, Two White) or 'WWW' (Three White). The girls were told that none of the four labels correctly described the contents of the box to which it was attached.

Each girl was told to draw two balls from her box, at which point the blindfold would be removed so that she could see the two balls in her hand and the label on the box assigned to her. She was given the task of trying to guess the color of the ball remaining in her box.

As each girl drew balls from her box, their colors were announced for all the girls to hear but the girls cold not see the labels on any other box other than their own.

The first girl, having drawn two black balls, looked at her label and announced: 'I know the color of the third ball!'

The second girl drew one white and one black ball, looked at her label and similarly stated: 'I too know the color of the third ball!'

The third girl withdrew two white balls, looked at her label, and said: 'I can't tell the color of the third ball.'

Finally, the fourth girl declared: 'I don't need to remove my blindfold or any balls from my box, and yet I know the color of all three of them. What's more, I know the color of the third ball in each of the other boxes, as well as the labels of each of the boxes that you have.'

The first three girls were amazed by the fourth girl's assertion and promptly challenged her. She proceeded to identify everything that she said she could.

Can you do the same?

<Puzzle retained as spoiler space>

I think so.

Girl one must have the box labelled BBB (in which case she knows the final ball is white) or the box labelled BBW (in which case she knows that the final ball is black). No other combinations would allow her to know the colour of the last ball.

Girl two must have either the box labelled BBW (final ball is white) or BWW (final ball is black).

Girl three must have either the box labelled BBB or BBW because if she had WWW she would know the final ball is black and if she had BWW she would know the final ball was white.

As BBB, BBW and BWW are shared between the first three girls, somehow, the 4th girl must have the box labelled WWW.

Girl 3 is the only one who can have the contents WWW.

Girl 2 is the only one who can have the box labelled BWW, so the contents are WWW and her final ball is white.

Now that girl 2 is eliminated girl 4 is the only one remaining who can have the contents BWW.

The only contents remaining for girl 1 is BBB, so her box must be labelled BBW, leaving girl 3 with the box labelled BBB.

So, girl 1 has the contents BBB, her box is labelled BBW and her final ball is black.

Girl 2 has the contents BBW, her box is labelled BWW and her final ball is white.

Girl 3 has the contents WWW, her box is labelled BBB and her final ball is white.

Girl 4 has the contents BWW and her box is labelled WWW, she's also a real smarty-pants to work all of that out in her head while blindfold.

I enjoyed the puzzle though.

Are we allowed to assume that the third girl wasn't a perfect logician? It seems to me that she had sufficient information to work out the colour of her third ball even if she couldn't actually manage it.

I think you're right. She has the box labelled BBB and she can therefore work out that girl 1 must have the box labelled BBW and the contents BBB.

Therefore girl 2 must have the contents BBW and the box labelled BWW.

So she knows that girl 4 must have the box labelled WWW, which must hold BWW, so she can determine that the final ball in her own box is white.

In article <203c97b955baa47aa92f53c80adf6d7e.61944

Just spent ten minutes typing long reply and answer and gravity newsreader just killed it

so here goes again

spoiler space

from girl 4 perspective

girl one (g1)

info= balls BLACK BLACK third - knows

possibilities label BBB

) )> The first three girls were amazed by the fourth girl's assertion and )> promptly challenged her. She proceeded to identify everything that she )> said she could. )> )> Can you do the same? ) ) Are we allowed to assume that the third girl wasn't a perfect logician? It ) seems to me that she had sufficient information to work out the colour of ) her third ball even if she couldn't actually manage it.

That's quite obvious, as the third girl has all the information the fourth girl has when she declares she doesn't know.

SaSW, Willem

Quite right, Duncan, unlike most puzzles of this kind.

original post as spoiler space

All four boxes have different labels, right?

[Assuming everyone is telling the truth and is a 'perfect logician'. If not then this because a problem of trans-Smullyan-esque difficulty :)]

The available boxes are BBB BBW BWW WWW.

Girl 1 draws BB and therefore has BBB or BBW. If she sees BBB she knows she has BBW and knows the third colour If she sees BBW she knows she has BBB and knows the third colour If she sees BWW or WWW she can't know whether she has BBW or BBB and so can't know the third colour. Therefore she sees BBx and has BBy where x = not y.

Girl 2 draws BW and therefore has BBW or BWW (She also knows from what she has heard that Girl 1 sees BBx and has BBy) If she sees BBB she knows (G1 sees BBW and has BB but nothing about her own box If she sees BBW she knows she has BWW and knows the third colour If she sees BWW she knows she has BBW and knows the third colour If she sees WWW she knows nothing Therefore she sees BpW and has BqW where p = not q

Girl 3 draws WW and therefore has BWW or WWW (She also knows from what she has heard that G1 sees BBx has BBy, G2 sees BpW has BqW) If she sees BBB she knows G1 sees BBW has BBB, that G2 sees BWW has BBW, but nothing about her own box If she sees BBW she knows G1 sees BBB has BBW, G2 sees BBW has BWW, knows she has WWW and knows the third colour If she sees BWW she knows G2 sees BBW has BWW, knows she has WWW, knows the third colour If she sees WWW she knows she has BWW, knows the third colour

Therefore the only way she doens't knows her own third colour is if she sees BBB

Girl 4 can do all the calculations we have done and comes to these conclusions: G1 sees BBW has BBB G2 sees BWW has BBW G3 sees BBB has xWW

Therefore she can conclude that G4 sees WWW (the only unused label) and has BWW or WWW; but not the latter since that would match her label; so

G3 sees BBB has WWW G4 sees WWW has BWW

SPOILER

The possible situations (2 per girl) are listed below:

Girl Balls Label Last drawn ball 1 BB BBB W BBW B 2 BW BBW W BWW B 3 WW BBB ? BBW ?

Using the above information only one combination does not cause a conflict:

Girl Balls Label Last drawn ball 1 BB BBW B 2 BW BWW B 3 WW BBB ?

The 4th girl knows her label is WWW, and so the 3rd girl's last ball must be white (otherwise she herself would have 3 white balls as on her label). Therefore she knows she has one black and two white balls, and can give the full details:

Girl Balls Label 1 BBB BBW 2 BBW BWW 3 WWW BBB 4 BWW WWW

Please reply to drgmayer at hotmail dot com

Here's the part I can't figure out.

If the 4th girl can tell, blindfolded that label #4 is WWW, then why can't the 3rd girl also tell that label #4 is WWW?

And if the 3rd girl can tell, then why can't she figure out that the balls in box #3 must be WWW?

Either the label is BBB and the contents is BBW, or vice versa.

Either the label is BWW and the contents is BBW, or vice versa.

The label is either BBB or BBW, and the contents is either BWW or WWW.

The first and third girls have used up the labels BBB and BBW, so the second girl's label can't be BBW; it's BWW, and her box contains BBW.

The first girl's box can't contain BBW, so it contains BBB, and her label is BBW. The third girl's label must be BBB, and so the fourth girl's label is WWW. The box with WWW cannot belong to the fourth girl; it must belong to the third, and the fourth's box has BWW.