> -----Original Message----- > From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On > Behalf Of maru dubshinki > Sent: Tuesday, October 10, 2006 10:41 PM > To: Killer Bs Discussion > Subject: Paradox, or, Breaking the mind of logic > > A while ago on #Wikipedia, I fell into a discussion with a fellow > editor. He posed me a question about the following riddle: > > Suppose there is an island with a number of natives on it. Each native > has either a red or a blue spot on their forehead. But they are not > allowed to indicate to each other or otherwise divine in any direct > observational fashion what the color of their particular spot might > be. One of the iron-clad customs of these indigenous persons is that > any native who deduces the color of their spot through logic must kill > themselves that midnight. > > Now, suppose further that of all the natives there, only two have blue > spots and all the rest have red spots. A outsider comes along (perhaps > he is an ignorant ethnographer), and truthfully mentions to the > natives that "At least one of you has a blue dot on your forehead." > > What will happen to the natives, and how long will it take?
My understanding, off the top of my head, is that the natives will be fine as long as they are not all together. After that, the first midnight, nothing will happen. Then, if the two people with blue dots meet after that midnight, they will kill themselves the following midnight. If the natives discover this before there are additional native, the rest will kill themselves by the following midnight. Dan M. > (Answer will be supplied below to further the cause of discussing the > question) > > > > > > spoiler > > Are you really sure you don't want to figure it out yourself? > > If you need a hint, you can always Google the problem. It's a classic > logic problem you know. > > > > > Well, if you're sure. > > Begin spoilers > > > > All the natives will eventually kill themselves. The precise number of > days is something like N+1 days. > The reason is that if either blue dot looks around, and sees the other > blue dot, they should believe that they are probably red (since most > natives are red), and thus there is only one blue dot, the other guy. > So they believe that the other blue dot will kill themselves that day. > Now, he won't (because there is another blue dot, and he is using the > same reasoning - to him, it's the *other* blue dot who should be > killing himself). On the second day, nobody will be dead, and so the > second blue dot must conclude that the reason for this is because they > themself are the second blue dot. Both will kill themselves. This same > reasoning can be generalized (mathematical induction?) for all numbers > 2 and above, since if 3 blue dots, they will wait to day 3 before all > the blues are dead, and so forth. If there is only one blue dot, then > they will kill themselves immediately, since there is at least one > blue dot, and they know that everybody except themself is red, thus > they must be the blue dot. > > Anyway, once the blues have killed themselves off, all the reds will > immediately commit suicide: they followed the blues' reasoning after > all, and know that all the blues are dead, which means that they are > red, and so since they know, they must kill themselves. > So that's that. Necessary background is over and done with. > > The problem that fellow editor posed me (once I'd solved the original > riddle) was this: in the case of 2 blue dots and a bunch of reds, each > blue dot *already* knows what the original stranger told them, that > there was at least one blue dot. They can see the other blue dot! So > it's quite obvious to them that there is at least one blue dot, and so > the stranger tells them absolutely nothing new - they already knew > what he told me, after all. What is not quite as obvious is that if > the stranger is removed, the 2 blues will exist in stasis; in the > first round, each will be waiting for the other to do something (their > situations remember are absolutely symmetrical), and so *every* round > they will be waiting for the other to do something, and of course > that means they never do anything. The reds are irrelevant since they > don't affect matters until the blues are gone. > > Now, the stranger appears to be absolutely useless, but nevertheless, > removed from the picture the whole thing breaks down in the case where > N = 2. What is the use of the useless stranger? > > ~maru > We put thirty spokes together and call it a wheel; > But it is on the space where there is nothing > That the usefulness of the wheel depends. > We turn clay to make a vessel; > But it is on the space where there is nothing > That the usefulness of the vessel depends. > _______________________________________________ > http://www.mccmedia.com/mailman/listinfo/brin-l _______________________________________________ http://www.mccmedia.com/mailman/listinfo/brin-l
