# Blue-eyes logic puzzle **URL:** \ **Category:** Factual Questions\ **Created:** [December 8, 2011, 5:31pm UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531 "2011-12-08T17:31:14Z")\ **Posts on this page:** 20\ **Page:** 22
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 14, 2011, 1:50am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/421 "2011-12-14T01:50:59Z")
> [@](#): > > Not really. > > The additional information is: Everyone now knows that everyone now knows that there’s at least one blue-eyed islander. You skipped about 98 “that everyone now knows” in there. ---
**Author:** ![emoticorpse](https://avatars.discourse-cdn.com/v4/letter/e/22d042/32.png) [@emoticorpse](https://boards.straightdope.com/u/emoticorpse)\ **Post date:** [December 14, 2011, 1:51am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/422 "2011-12-14T01:51:01Z")
maybe we’re all wrong on this one? might be a puzzle in itself what the real value was of the guru’s statement? ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [December 14, 2011, 1:52am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/423 "2011-12-14T01:52:42Z")
> [@Galileo](#): > > The additional information is: Everyone now knows that everyone now knows that there’s at least one blue-eyed islander. Though “at least one” and “not zero” are equivalent, I think your wording may be better. The critical point is that “everyone knows that everyone knows…”. ---
**Author:** ![Greg\_Charles](https://avatars.discourse-cdn.com/v4/letter/g/839c29/32.png) [@Greg\_Charles](https://boards.straightdope.com/u/Greg_Charles)\ **Post date:** [December 14, 2011, 2:15am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/424 "2011-12-14T02:15:18Z")
> [@Frylock](#): > > Do you think that if only three had had blue eyes, then the information that at least one of them does wouldn’t have been news to anyone? > > If so, do you think, nevertheless, that in the three-blue-eyed-people scenario, they would have reasoned their way to leaving on the third day? > > If not, please go back in the thread to the start of my conversation with **Emoticorpse** and follow the reasoning that we illustrate together and tell me what you think. Yes, I do think the blue-eyed people would have reasoned their way to leaving the third day. I don’t think you read my question. > [@Ponderoid](#): > > No, I already thought of that too, and ruled it out, see [post 252](http://boards.straightdope.com/sdmb/showpost.php?p=14554685&postcount=252). I’m finding your stream-of-consciousness a bit hard to follow, but you might be on to something. I’m starting to think it might have to do with there being an equal number of brown-eyed to blue-eyed. If there were more, they could escape the island as well. I’ll try to logic out that theory and see if I’m right. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [December 14, 2011, 2:25am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/425 "2011-12-14T02:25:36Z")
> [@Greg\_Charles](#): > > Yes, I do think the blue-eyed people would have reasoned their way to leaving the third day. I don’t think you read my question. From this: > [@](#): > > However, I have to agree that if 100 people have blue eyes, then the information that at least one of them does isn’t news to anyone. I took you to be intending to imply that you don’t think the reasoning we’ve described works for the 100-blue-eyed-people case. If that’s not what you meant to imply, then I’m not sure what you made this statement for. > [@](#): > > I’m finding your stream-of-consciousness a bit hard to follow, but you might be on to something. I’m starting to think it might have to do with there being an equal number of brown-eyed to blue-eyed. If there were more, they could escape the island as well. I’ll try to logic out that theory and see if I’m right. The brown-eyed islanders never escape because, though they know now (after the mass exodus) that they don’t have blue eyes, they still have no basis for determining what color eyes they _do_ have. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [December 14, 2011, 2:27am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/426 "2011-12-14T02:27:43Z")
> [@emoticorpse](#): > > oh well i tried i seriously hope to get as smart as the other people on this site. I just started though joined straightdope like a week ago You basically get it AFAICT. Knowing why the blue-eyed ones leave is different from knowing exactly how the Guru has contributed any information. Hopefully our conversation, and my recent “quote-box-within-quote-box-within-quote-box etc” post, illustrate that the information is just this: “The guru has said that she sees at least one blue eyed islander.” That’s something no one knew before (because it wasn’t true) and now that they know it, they can commence to logickin’. ---
**Author:** ![Galileo](https://avatars.discourse-cdn.com/v4/letter/g/85e7bf/32.png) [@Galileo](https://boards.straightdope.com/u/Galileo)\ **Post date:** [December 14, 2011, 2:28am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/427 "2011-12-14T02:28:18Z")
> [@Chronos](#): > > You skipped about 98 “that everyone now knows” in there. I don’t think so. The information conveyed by the Guru’s statement does not vary with the number of blue-eyed islanders. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 14, 2011, 2:36am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/428 "2011-12-14T02:36:31Z")
Well, really, you skipped an infinite number of layers, but it’s the first n (where n is the number of blue-eyed islanders) that are the important ones. But “Everyone now knows that everyone now knows that there’s at least one blue-eyed islander.”, as you put it, is not sufficient unless we’re in a very small case. ---
**Author:** ![Galileo](https://avatars.discourse-cdn.com/v4/letter/g/85e7bf/32.png) [@Galileo](https://boards.straightdope.com/u/Galileo)\ **Post date:** [December 14, 2011, 2:41am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/429 "2011-12-14T02:41:29Z")
I am not talking about the chain of induction. I’m talking about the additional information that was conveyed by the Guru’s announcement. The value of n was not conveyed by the Guru, nor was it necessary that it be conveyed – it was already known by all the islanders and they knew that everyone knew it. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [December 14, 2011, 2:47am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/430 "2011-12-14T02:47:33Z")
> [@Galileo](#): > > I am not talking about the chain of induction. I’m talking about the additional information that was conveyed by the Guru’s announcement. The value of n was not conveyed by the Guru, nor was it necessary that it be conveyed – it was already known by all the islanders and they knew that everyone knew it. The more I tihnk about it the more I tihnk the right way to put the new information is just as follows: > [@The New Information](#): > > The Guru says there is at least one blue eyed islander. If there’s any distinction between information provided and inferences drawn from that information, then the above is the information provided. All the everybody knows that everybody knows etc stuff is inference drawn from that information. ---
**Author:** ![Galileo](https://avatars.discourse-cdn.com/v4/letter/g/85e7bf/32.png) [@Galileo](https://boards.straightdope.com/u/Galileo)\ **Post date:** [December 14, 2011, 3:15am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/431 "2011-12-14T03:15:53Z")
> [@Frylock](#): > > If there’s any distinction between information provided and inferences drawn from that information, then the above is the information provided. Fair enough. But the XKCD puzzle already says this: > [@](#): > > [The Guru is] simply saying “I count at least one blue-eyed person on this island who isn’t me.” Perhaps it’s more precise to say that the first inference drawn from the information is that everyone now knows that everyone now knows that there’s at least one person with blue eyes. In other words, there is new common knowledge. ---
**Author:** ![drewtwo99](https://avatars.discourse-cdn.com/v4/letter/d/d07c76/32.png) [@drewtwo99](https://boards.straightdope.com/u/drewtwo99)\ **Post date:** [December 14, 2011, 3:38am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/432 "2011-12-14T03:38:11Z")
Is anyone ever going to tell me if I got the lightning bolt questions right? I’m dying to know! ---
**Author:** ![AHunter3](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ahunter3/32/368_2.png) [@AHunter3](https://boards.straightdope.com/u/AHunter3)\ **Post date:** [December 14, 2011, 3:58am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/433 "2011-12-14T03:58:39Z")
I would say a blue-eyed person leaves. A logical blue-eyed person is looking at the guru (green-eyes) and one other person. If the other person has brown eyes, this logical blue-eyed person would be on the boat to leave the island before the final syllable finishes echoing. Because the guru could only be seeing HER eyes as blue if the other person has brown eyes. Let’s say, however, that the blue-eyed person is not in the enviable position of staring at a brown-eyes person and a green-eyed guru. Instead, she sees another blue-eyed person. Well, she says to herself, IF, hypothetically speaking, yon blue-eyed person were looking at my brown eyes and hearing Guru say there’s a blue-eyed person present, my colleague here would be there, halfway up the boarding ramp. So let’s say that the other blue-eyed person is not hopping up to do that. So she cogitates. If I had brown eyes my blue-eyed companion here would have figured this out. We ain’t dumb. We’re all pretty logical here. Therefore the absence of quick conclusions on the part of my colleague means I, too, must have blue eyes. “I have blue eyes!” ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [December 14, 2011, 4:41am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/434 "2011-12-14T04:41:47Z")
Without going back through all the posts, I’ll take another shot at this. In order to be able to deduce what color your eyes are, you need the following information: Would a single blue-eyed person be able to figure out what color their eyes were with the information they have? Looking around and seeing blue-eyed people doesn’t provide enough information for this. But the Guru’s pronouncement does. This is the key. ---
**Author:** ![Carmady](https://avatars.discourse-cdn.com/v4/letter/c/f04885/32.png) [@Carmady](https://boards.straightdope.com/u/Carmady)\ **Post date:** [December 14, 2011, 4:43am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/435 "2011-12-14T04:43:54Z")
> [@drewtwo99](#): > > Is anyone ever going to tell me if I got the lightning bolt questions right? I’m dying to know! Half right. If lightning strikes a blue eyed islander on day 2, all the rest still leave on day 100 like they would have if no lightning struck. But if lightning strikes right after the announcement, it nullifies the announcement, and nobody ever leaves. ---
**Author:** ![nivlac](https://avatars.discourse-cdn.com/v4/letter/n/3bc359/32.png) [@nivlac](https://boards.straightdope.com/u/nivlac)\ **Post date:** [December 14, 2011, 5:10am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/436 "2011-12-14T05:10:58Z")
> [@emoticorpse](#): > > maybe we’re all wrong on this one? might be a puzzle in itself what the real value was of the guru’s statement? The original puzzle simply asks “Who leaves and on what night?” That’s been answered hundreds of posts ago and can be proven mathematically. Then the discussion veered off to “What new information did the Guru provide?” Interesting, but for those who haven’t figured out the solution, just muddies up the waters for them because the key is not whether the information is new or not, but whether it becomes [common knowledge](http://en.wikipedia.org/wiki/Common_knowledge_%28logic%29). ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [December 14, 2011, 5:16am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/437 "2011-12-14T05:16:14Z")
> [@AHunter3](#): > > … this logical blue-eyed person would be on the boat to leave the island before the final syllable finishes echoing. It would have to be a long echo: the Guru speaks at noon, but the ferry doesn’t shows up until midnight. ---
**Author:** ![Kevbo](https://avatars.discourse-cdn.com/v4/letter/k/e47774/32.png) [@Kevbo](https://boards.straightdope.com/u/Kevbo)\ **Post date:** [December 14, 2011, 6:07am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/438 "2011-12-14T06:07:09Z")
I have been lurking about this thread since the post count was in the single digits. I haven’t posted because my argument was well presented by others. The inductive explanation fails for me for the chief reason that others have posted: Every single day before the guru’s pronouncement, the facts contained in the pronouncement were known to every single islander. Well I finally cracked the nut. It works in spite of the fact that the guru didn’t provide any new information. Before the guru’s pronouncement, the question every islander was trying to answer was “what color are my eyes?” Since there was no limit to the possible colors, this question could never be answered. The key is that the guru added no information related to answering the original question the islanders needed to answer. What, in fact, the guru did was to change the question! The guru’s pronouncement changed the question to “Are my eyes blue or not?” 100 of the islanders now know that this question can be answered on day 99 or day 100 after the pronouncement, and the other 100 now know they will know the answer on day 100 or 101. Note that this question has a yes/no answer, instead of the infinite number of possible answers to “what color are my eyes?” There is no “I know that he knows that she knows the he knows…” required at all. I concede that this inductive argument might be valid. I honestly can’t follow it, and I for sure can’t get over the fact that something that was previously known to every islander somehow gets the ball rolling just by being spoken out loud. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 14, 2011, 7:23am UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/439 "2011-12-14T07:23:54Z")
We all happen to already know your credit card number, **kevbo**. We’ve known it for years. If someone were to suddenly post your credit card number, well, they’d be posting information we all already know. But it _would_ change something. Because up until quite recently, you haven’t been aware that we all know your credit card number. Of course, once you saw your credit card number posted publicly, you’d have no doubt that everyone now knows your credit card number. Even though the information posted would be information we all already knew, and in particular information you already knew, it would still change what you knew about what the rest of us knew. What’s going on _is_ that same phenomenon, only with more layers of it. It’s possible to publicly say something that everyone already knows, and in so doing, change higher-order knowledge… ---
**Author:** ![Ximenean](https://avatars.discourse-cdn.com/v4/letter/x/aca169/32.png) [@Ximenean](https://boards.straightdope.com/u/Ximenean)\ **Post date:** [December 14, 2011, 12:35pm UTC](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531/440 "2011-12-14T12:35:41Z")
> [@Kevbo](#): > > Well I finally cracked the nut. It works in spite of the fact that the guru didn’t provide any new information. > > Before the guru’s pronouncement, the question every islander was trying to answer was “what color are my eyes?” Since there was no limit to the possible colors, this question could never be answered. > > The key is that the guru added no information related to answering the original question the islanders needed to answer. What, in fact, the guru did was to change the question! But they \*could \*have been wondering “do I have blue eyes?” all along, and yet they would never have been able to answer it, and no blue-eyed people would ever leave. There’s no getting round it - the Guru must be supplying some new information. [Previous page](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531.md?page=21) [Next page](https://boards.straightdope.com/t/blue-eyes-logic-puzzle/605531.md?page=23)