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Logic Puzzle Advertising

There is an accounting firm that has been using puzzles of various types in their ads. Today on the train into work, I saw one of their ads, with a logic puzzle. I am posting the puzzle here and inviting readers to comment with their solutions and reasoning behind the solution. Comments will be screened until at least Wednesday (and maybe as long as Friday).

The Puzzle:

Three supermodels are invited into a swanky and exclusive Fifth Avenue shop. They are shown a shelf upon which are five pashminas - three in deep cerulean and two in saffron. All three are then blindfolded, and one pashmina is draped over each. The remaining two pashminas are returned to the shelf. The models are lined up so that none of them can see the shelf.

The third supermodel's blindfold is removed, and she is asked whether she can tell, by looking only at the other two supermodels, what color pashmina she is wearing. She replies that she cannot.

The second supermodel's blindfold is removed, and she is asked whether she can tell, by looking only at the first supermodel (and, obviously, having heard the third supermodel's reply), what color pashmina she is wearing. She, too, replies that she cannot.

The first supermodel, before her blindfold can be removed, says "I am wearing a ___________ pashmina - may I keep it?".

What color pashmina was the first supermodel wearing, and how do you know?

Comments
The Answer

The first supermodel was wearing a deep cerulean pashmina.

Recall that there were only two saffron pashminas. When the third supermodel was unable to determine what color pashmina she was wearing, it meant that she could not have seen two saffron pashminas, or she would have immediately known that hers was deep cerulean. Since she did not know, she must have seen either two deep cerulean pashminas, or one saffron and one deep cerulean.

When the second supermodel looked at the first, and knew that the third could not determine what color she was wearing, the second supermodel correctly concluded that the third could not have seen two saffron pashminas. So, if the second had seen a saffron pashmina, she would have known immediately that she was wearing a deep cerulean pashmina, as that would have been the only possibility that would not have allowed the third supermodel to know what color she was wearing. Since the second supermodel did not know what color she was wearing, she could not have seen a saffron pashmina, and thus must have seen a deep cerulean pashmina.

The first supermodel, having heard both of the others say that they could not identify their respective pashminas, used the above logic to immediately know - without knowing the color of either of the other pashminas - that she was wearing a deep cerulean pashmina.

The third supermodel sees two other models wearing blue pashminas. Either they are both wearing blue ones, or one is wearing blue and the other one saffron. Either way, she doesn't have enough information to know whether she is wearing the remaining blue one or the remaining saffron one.

The second model has heard the third one's answer, and sees the that the first one is wearing blue. Again, whether it is blue or saffron, she doesn't have enough information to tell what she is wearing.

The first model now knows that she and the second model were both wearing blue pashminas. Therefore, she is wearing a blue one. Let her keep it ;-D

Edit: I hadn't finished typing before you posted the answer :-(





Edited at 2009-03-23 12:50 pm (UTC)

The first supermodel is a ringer.

Oh. and she's wearing a cerulean pashmina.

What the 3rd model might see:

1 2 3 Model
C C ? Color
C S ?
S C ?
S S C

She couldn't figure it out, eliminating the last possibility. Second model has choices:

1 2 3 Model
C ? x Color (first two from above combined)
S C x

She couldn't figure it out either, eliminating the last possibility. The only two possibilities left have the first model wearing cerulean.

Here's my reasoning, which is probably wrong.

If supermodel three saw two saffron pashminas, she would know hers was cerulean. Since she didn’t know, she saw one of the following sets:
-- Two ceruleans
-- One cerulean and one saffron

The possible combinations are:
-- SM 1 Cerulean, SM 2 Cerulean
-- SM 1 Cerulean, SM 2 Saffron
-- SM 1 Saffron, SM 2 Cerulean

If supermodel two saw one saffron pashmina, she would know hers was cerulean. Since she didn’t know, supermodel 1’s pashmina has to be cerulean.

First is wearing a deep cerulean one.

Third can see either two cerulean or one cerulean and one saffron. (If she could see two saffron then she'd know she had a cerulean.)

Second can, likewise, see either two cerulean or one cerulean and one saffron.

If third can see two cerulean and second can see two cerulean then first must be wearing cerulean.

If third can see two cerulean and second can see one saffron and one cerulean, then third must be wearing the saffron, so first is wearing cerulean.

Likewise swapping third and second.

You've got the correct answer, but your reasoning is incorrect; you're assuming that each model can see both of the other two, but while #3 can see both #1 and #2, #2 can only see #1.

Ah. Missed that bit. Or forgot it when I was writing out the explanation. Accounts for the subsequent uncertainty I was feeling.

If SM1 and SM2 are wearing safron then SM3 would know that she is wearing celurean. But this is not the case,

So we can conclude that SM1 and SM2 are either both wearing celurean, or safron and celurean.

If SM1 was wearing safron, then SM2 has to be wearing celurean, since both of them can't be wearing safron. But from SM2 we can conclude that SM1 is not wearing safron.

SM1 must therefor be wearing celurean.

Are you sure that's all the information? It does not appear solvable given the provided information.

Yes. You know everything you need to in order to determine the color of model #1's pashmina. You do not have sufficient information to determine the color of either of the other pashminas, but that's not part of the puzzle.

If #3 saw red scarves on both #1 and #2, she would know the remaining 3 (including her own) are blue. Since she can't tell the color of her own scarf, at least one of #1 and #2 has a blue scarf.

If #2 saw a red scarf on #1, she would know, having heard #3, that her own scarf is blue. Since she can't tell the color of her own scarf, #1's scarf must be blue.

okay. if #3 saw two saffron, she would know hers was cerulean. therefore she sees CC or SC or CS. if #2 saw S she'd know hers was C because otherwise #3 would have seen SS. therefore #2 saw C and #1 knows what colour her pashmina is.

If model 1 and 2 both bore saffron, then model 3 would know that hers could only be cerulean, as all the saffron would be accounted for.

That is not the case.

If model 2 sees saffron on model 1, she would know that she has cerulean one on, otherwise model 3 would have answered differently.

This leaves cerulean as the only color model 1 could be wearing.

The first supermodel is wearing a cerulean pashmina.

I'll put my reasoning in a response so it can be screened separately.

Well, three supermodels, two colors of pashminas (pashminae?). Eight possibilities. So, let's just elaborate the possibilities:

Possibility #Supermodel 1Supermodel 2Supermodel 3
0SaffronSaffronSaffron
1SaffronSaffronCerulean
2SaffronCeruleanSaffron
3SaffronCeruleanCerulean
4CeruleanSaffronSaffron
5CeruleanSaffronCerulean
6CeruleanCeruleanSaffron
7CeruleanCeruleanCerulean


Well, no, there are only seven possibilities. Possibility #0 isn't valid, because all three supermodels are wearing saffron pashminas, but there are only two saffron pashminas to go around.

So, let's look at possibility 1. Supermodel three looks at supermodels one and two, sees that both saffron pashminas are accounted for, knows she must be wearing a cerulean one, and would answer "yes" to her question. However, the problem states she answered "no", so the answer can't be possibility 1.

Since supermodel three said "no", supermodel two knows that she and supermodel one are together wearing at most one of the two saffron pashminas: either supermodel two is wearing one, or supermodel one is wearing one, or neither supermodel are.

So, in possibilities 2 and 3, supermodel two looks at supermodel one, sees a saffron pashmina, knows she must be wearing a cerulean one, and would answer "yes" to her question. However, the problem states she answered "no", so the answer can't be possibilities 2 or 3.

In all remaining possibilities (4 through 7), supermodel three looked at supermodels one and two, saw at most one saffron pashmina, so answered "no". Also, supermodel two looked at supermodel one, did not see a saffron pashmina, could not tell what color pashmina she was wearing, so answered "no".

However, in all remaining possibilities (4 through 7), supermodel one was wearing a cerulean pashmina. So, that is what she must have been wearing.

QED.

(Anonymous)
1st pashmina

1st pashmina is cerulean. Because in order for 3rd to say she doesn't know then at lease ONE of the 1st two must be cerulean - because if the 1st two were saffron then since there are only two saffron then 3rd supermodel would know she herself is wearing cerulean. Now only way 2nd supermodel could know her own color is if 1st supermodel is saffron (in which case, based on 3rd model's reply she would know her own must be blue). But 2nd model also doesn't know, therfore only possibility is that 1st is wearing cerulean.

A variant of the old Three Hats (or Three Philosophers) puzzle (though somewhat different from the also well known "Hat problem").

http://www.greylabyrinth.com/puzzle/puzzle007
http://tierneylab.blogs.nytimes.com/2009/03/16/the-puzzle-of-the-3-hats/?partner=TOPIXNEWS&ei=5099


Yes, there are several variants of this puzzle. And probably of every other logic puzzle that either of us is likely to encounter...