There aren’t any days that don’t contribute useful information. Every day raises the base, and without each day’s raise, the next day’s isn’t possible. You can’t raise the base case to 97 before you raise it to 96, or to 96 before you raise it to 95, or to 95 before…
This doesn’t depend on whether or not they want to get off the island, but on the common knowledge they use to deduce their eye color.
The problem with common knowledge is that they already have it. Everyone can already see 99 people with blue eyes. They know that nobody will leave on day 1 … 97. That’s why the problem seems so artificial. It prompts a search for ‘faster’ solutions.
The browns and blues are in the same situation, thus it doesn’t matter which is being counted; the blues or the browns.
Why 97? Why did you stop at 96? If I see 99, I know nobody is going to leave on day 96, 97, and 98 as well. I don’t know why you stopped at 96.
Now you are getting close to asking the right question.
Of course if you would answer one honestly, you could get there a lot sooner (like 3-4 pages ago).
There’s some number which makes sense. I’ll take James’ word that it’s 97 because he’s been thinking about it more than me.
I’m not asking ‘which number makes sense’ though.
Your reasoning involved knowing what days nobody is going to leave on. If that’s the route, you end at 98, not 96.
If you end at 96, you must be using some criteria other than what days we know nobody is going to leave on.
What is it?
I don’t think you’re one to talk about honest questions in this thread.
“What is an eye?”
This was your question.
My argument that I presented with a few pages back debunked the whole “If there’s 100, you can start with 97” argument.
The reasoning is that, no matter what number there are, his idea is that you always start with 2 less than the person sees.
There are 2 problems with this:
No you didn’t (again).
What you debunked (again) was merely that a person could not just take “2 less than he sees”.
As always, you rush to conclude your own success.
That’s what the “start with 97” argument WAS. “Take 2 less than he sees” = “start with 97”. That’s what I debunked. Yes. The only solution you offered which involved starting from 97 was debunked. So unless there’s a new one, I don’t see why we’re still thinking it’s a good idea to start from 97.
No it wasn’t “what the 97 argument was”. Again, merely trying to claim a victory.
The argument was that IF EVERYONE STARTS WITH THE SAME NUMBER, IT WILL ALWAYS WORK.
The 97 was an example. When you started questioning it, I rushed, mistakenly, into thinking “well just take 2 less than you see”. Okay, it isn’t THAT simple. You can’t just take 2 less than you see.
Urgh.
The whole reason this 97-nonsense (or 98/96/99 whatever) doesn’t work has already been covered.
There are 161700 combinations of 97 blues amongst 100. Just because everyone can know that everyone can know that everyone sees 97, doesn’t mean anyone has any clue about which combination is the one that everyone is hypothesising about when they are thinking “well IF there were the only those 97, they could leave on the 1st day”, as in the proposed 97 “solution”.
As such, nothing is gained as exhaustive knowledge when nobody leaves on the 1st day, so there is nothing firm on which to continue to deduce about 98 on the next day, and so on. It also falls foul to the transfer of the knowledge of 100 if there were only 100 onto hypotheses about the knowledge of 97 if there were only 97 (i.e. out of context). Also already covered.
When the correct solution has built up knowledge about whether 97 leave on the 97th day, the number 97 is actually founded on something definite and tested about 96, 95, 94 and so on, right down to 1, and the solid irrefutable logic that if there was 1 blue-eyed islander and he saw no blue-eyed islanders, and the Guru said her words, he would leave on the first day. If there were 2, and each saw 1, + Guru words, and it was tested that nobody left after the first day, then they both know they each have blue eyes too, and so on, with each day being absolutely necessary in order to have a firm, undeniable logical backing to what 97 would do - which is shown can ONLY be definitely and exhaustively known on the 97th day, just like what 100 would do can only be known on the 100th day when they can finally all leave with this definite and exhaustive knowledge.
If they had all started their count from 97, would it have worked?
If they had all started their count from 50, would it have worked?
If they had all started their count from 01, would it have worked?
If the guru said “I see 97 with blue eyes”, it would work.
If she said “I see 50 with blue eyes” it would work.
If she said “I see 01 with blue eyes” it would work.
If everyone just picked a number out of thin air, even if they all happened to pick the same number, it would not work.
Not without the Guru imparting information that would go against the logically deduced expected result when going only by what can be seen for definite about the eye colour of all others, and thereby revealing information about the one unknown to any one blue-eyed islander: their own eye colour. THAT is the key. They are the ONLY two possibilities.
As Carleas says, if the Guru said she can see 97 people who have blue eyes, and they can start their count from 97, if the Guru said she can see 50, they can start from 50, just like they can start from one if and only if the Guru says she can see someone who has blue eyes.
Really?
Is the number, the guru says, some secret code which allows them to skip days?
It’s a “secret code” in that it’s able to give extra information to the blue-eyed islanders under special key circumstances in the deductive process, even though at other points it just seems obvious to anyone.
And yes, the amount of islanders she refers to makes a difference that could skip days. Consider:
If she said she saw 100 blue eyed islanders, each blue would see 99 and deduce the 100th must be themselves. They all leave at the first opportunity. Each brown would already see 100 blues and think that no more information was added to what they could already see AND deduce.
If she said she saw 99 blues, and there were only 99 blues, they would leave by the same logic.
If she said she saw 99 blues, and there were 100 blues, each blue would be waiting for the 99 that they see to leave on the first day. They don’t because they’re all expecting others to leave and not themselves. They would each have deduced that there were either the 99 blues that they could see, or that there were 100 and each of them were the 100th after the 99 they each counted. Therefore since it’s not the former, known and tested after the first day, they all leave on the 2nd.
If she said 98 and there were 100, the same logic continues in the way we should be used to by now. She can say any number and make it solvable, just as long as the number is less than or equal to the actual total amount of islanders with the eye colour mentioned by the Guru, and as long as the number is 1 or more. The same logic goes, it just makes the process either faster or slower.
The critical things to note are that the Guru’s information conflicts with what is known just from looking at some point in the deductive process, and that the process MUST only consider what is definitely known for sure, and deductively exploring SOLELY within those confines in order to avoid risky assumptions like the alternative attempts mentioned so far, and that no information is taken out of context, such as the information known by 100 blues being used to deduce about what would happen if there were 1 blue knowing only what 1 blue would know. Once you get all these things, you should be home free to accepting the correct solution. Well done if you can, this is apparently very hard to do for some.
Really?
Then you should be able to give such an example (as requested pages ago).
Please tell us one number between 0 and 99 such that “even if everyone picked that same number”, it would still not work to allow them to discover their actual eye color.
James, I gave you an example (pages ago): any number between 0 and 99. Take 98 because it’s easy to work with: if every islander magicallyhad 98 in their head, they would be able to conclude exactly nothing about their eye color from it, no matter how many days elapsed.
There is no deductive process between “I have number 98 in my head” and “I have blue eyes”. The only way any deduction happens is if the number means something, and they know it means something.