perplexus.info :: Logic : Three women, three numbers

Three women, three numbers (Posted on 2005-12-15)

Three women were seated around a table. After blindfolded, a numbered disc was pasted on each of their foreheads. The women were truthfully told "Each of you has either a 1, a 2 or a 3 on your forehead, and the sum of your numbers is either 6 or 7".

After the blindfolds were removed, each woman in turn was asked to name the number on her forehead without seeing it. The question was repeated until only one woman failed to name the number on her forehead.

When it was logically possible to name the number on her forehead, each woman did so; when it was not logically possible to name the number on her forehead, she would said "I don愒 know my number", and waited until the question was repeated to her next time around.

Each woman had a 2 on her forehead. Which woman failed to name her number?

Submitted by pcbouhid

Rating: 3.3750 (8 votes)

Solution: (Hide)

When a woman sees two 2愀 on the other two women, she knows she had a 2 or a 3 on her forehead (the sum is 6 or 7).

In the first round of asking, each woman reasoned like this :

First woman: "I have a 2 or a 3, so I don愒 know my number".

Second woman: "I have a 2 or a 3. If I have a 2, the first woman wouldn愒 know whether she had a 2 or a 3. If I have a 3, first woman wouldn愒 know whether she had a 1 or a 2. So I don愒 know whether I have a 2 or a 3."

Third woman: "I have a 2 or a 3. If I have a 2, first woman wouldn愒 know whether she had a 2 or a 3 and second woman wouldn愒 know whether she had a 2 or 3 (see above). If I have a 3, first woman wouldn愒 know whether she had a 1 or a 2 and second woman wouldn愒 know whether she had a 1 or a 2; second woman would reason: "if I have a 1, first woman wouldn愒 know whether he had a 2 or a 3 and if I have a 2, first woman wouldn愒 know whether she had a 1 or a 2." So I don愒 know whether I have a 2 or a 3.

First woman: I have a 2 or a 3. If I have a 3, second woman would know she had a 2; second woman would know she didn愒 have a 1 because otherwise I would have known I had a 3; second woman would know she didn愒 have a 3 because the total is 6 or 7. Second woman didn愒 know she had a 2; so I have a 2.

Second woman: I have a 2 or a 3. If I have a 3, third woman would know she had a 2; third woman would know she didn愒 have a 1 because otherwise I would have know I had a 3; third woman would know she didn愒 have a 3 because the total is 6 or 7. Third woman didn愒 know she had a 2; so I have a 2.

Third woman: I have a 2 or a 3. If I have a 2, each of first and second women would know the second time around she had a 2 (see above). If I have a 3, first woman would know she had a 2; first woman would know the first time around she had a 1 or a 2; first woman would know the second time around she didn愒 have a 1 because otherwise I would have known I had a 3. If I have a 3, second woman would know she had a 2; second woman would know the first time around she had a 1 or 2 (see third woman愀 reasoning first time around); second woman would know the second time around she didn愒 have a 1 because otherwise I would have known I had a 3. So I still don愒 know whether I have a 2 or a 3.

Then the third woman asked failed to name her number, since at this point no more information can be obtained by she.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-04-14 22:35:04
**	Charlotte	2006-02-26 09:30:56
solution	matcotton	2006-01-17 04:35:12
Re: An Answer (The correct one?)	George Wade	2005-12-26 11:14:03
solution	surya	2005-12-21 05:07:30
Response to Gadsen	Gordon Steel	2005-12-17 14:10:03
re: Solution (I think) - Any Comments?	Gadsden	2005-12-17 10:19:23
Solution - Correction to Prior Posting	Gordon Steel	2005-12-16 17:21:08
Solution	Gordon Steel	2005-12-16 17:13:15
No Subject	DYLAN	2005-12-16 16:19:35
Hmmm...	Nick	2005-12-16 14:31:01
formatting fixed in solution	Wallace North	2005-12-16 10:51:30
Maybe this is it (cont.)	Kevin Beggs	2005-12-16 05:35:19
Maybe this is it	Kevin Beggs	2005-12-15 22:39:30
Systematic solution	Tristan	2005-12-15 21:28:03
Solution (I think)	Gadsden	2005-12-15 19:41:42
The correct solution (I think)	Gadsden	2005-12-15 19:30:56
re: Answer (i hope) - difficult to read	pcbouhid	2005-12-15 19:00:55
Answer (i hope)	Wallace North	2005-12-15 17:30:51
re: Before I Check (spoiler)	pcbouhid	2005-12-15 15:07:49
Before I Check (spoiler)	Vernon Lewis	2005-12-15 14:14:49
Solution	Gamer	2005-12-15 13:44:25
re: Solution - not quite	goFish	2005-12-15 13:25:35
Solution	Gamer	2005-12-15 12:59:16
re(2): An Answer?	goFish	2005-12-15 11:28:57
re: An Answer	pcbouhid	2005-12-15 10:45:07
An Answer	goFish	2005-12-15 10:39:43
a guess no proof yet	Wallace North	2005-12-15 10:12:31

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