Yes, I believe Dan's reply to Gene has it exactly right. In order for the puzzle to work, everyone must hear the traveler's statement. Gene comments that the information provided by the traveler was already known to everybody. I find this the most fascinating aspect of the puzzle. For it is true that everybody already knows that "Some people in this village have blue dots", since everybody can see at least nine blue dots in the village. The traveler, however, must be providing new information; otherwise, the deductions that play out as a consequence would have played out long ago. And, as Dan points out, the information that the traveler is unwittingly providing is that now everybody knows that everybody knows that everybody knows . . . that some people have blue dots. This is clearest in the case of a village with just two blue dots, B1 and B2: prior to the utterance by the traveler, B1 did not know that B2 knew that there were some blue dots. JSS Dan Hoey wrote:
Eugene Salamin <gene_salamin@yahoo.com> wrote:
--- Joshua Singer <singer@stanford.edu> wrote:
As it was originally told to me . . .
A traveler passes through a small village, each of whose inhabitants has a single colored dot .... ... As the traveler passes through, he casually remarks, "Some people in this village have blue dots". Ten days later, ....
This can't be right. The information provided by the traveler was already known to everybody. In the case of the problem of the unfaithful wives, the king issues an edict, which starts the clock ticking.
I believe the phrase "he casually remarks" should read "he publically announces," meaning that he states it in a way that (1) everyone hears and believes him, (2) everyone knows that (1) occurs, (3) everyone knows that (2) occurs, (4) everyone knows that (3) occurs, etc. all at the same time.
It takes about N such pieces of information to make the clock tick for N days.
Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun