The women are named A, B and C reflecting the order in which they answer. The situation is described (2,2,2) implying that A, B and C are respectively wearing the number 2 on their foreheads. A1 labels A’s 1^st answer.<o:p></o:p>

Each woman knows that she is wearing 2 or 3 (i.e. not 1)<o:p></o:p>

Answers:<o:p></o:p>

A1 – DK (don’t know)<o:p></o:p>

B1 - DK<o:p></o:p>

now A knows that she is wearing 2. <o:p></o:p>

Why?<o:p></o:p>

Consider A’s only alternative situation (3,2,2): A's answer, seeing B and C both wearing 2, is DK. However this answer informs B that she (B) is wearing 2. B can't be wearing 3 (because the sum of the numbers would then be 8) and she can't be wearing 1 (because A would have seen B:1 & C:2 and would have known that she was wearing 3).<o:p></o:p>

C1 - DK<o:p></o:p>

Why?<o:p></o:p>

Consider C’s only alternative situation (2,2,3): A’s answer, seeing (B:2, C:3) is DK (she could be wearing 1 or 2). B’s answer is then also DK, for A would have answered the same had B been wearing either 1 or 2.  <o:p></o:p>

Thus after hearing A&B both answer DK, and seeing them both wearing 2, C is unable to distinguish between situations (2,2,2) and (2,2,3).<o:p></o:p>

A2: I’m wearing 2 (for explanation see above)<o:p></o:p>

B2: I’m wearing 2<o:p></o:p>

How does B know she’s wearing 2?<o:p></o:p>

Consider B’s only alternative situation (2,3,2): A’s 1^st answer (seeing 3&2) would have been DK. B seeing 2&2 answers DK. B’s answer tells A&C that they’re both wearing 2. (Had either A or C been wearing 1, B would have had to have answered 3.) Thus C’s first answer (C1) would have been that she was wearing 2 and not DK.<o:p></o:p>