The Answer 39
The first thing that should become obvious is that there generally won't be a single question that will satisfy the criteria of an accurate survey, non-anonymously carried out, that gets a gazelle to confess they have Geo-orgosis, while also containing no information that proves they have it. So you realise it must be a two-step process.
You should also realise that it is impossible to avoid asking at least one gazelle if they have the virus.
It therefore sounds impossible, since no-one can be surveyed anonymously. But there IS a loophole. While it's true you cannot keep the answer a secret...the real genius lies in the realisation that what needs to be kept secret is the QUESTION.
And the best way to that is to ask one of two questions, and leave it up to chance regarding which one gets asked, to create plausible deniability for the surveyed gazelles. So let's introduce a random element, such as tossing a coin.
- Get your surveyed gazelle to toss a coin. If heads, ask if they have Geo-orgosis. If tails, get them to toss the coin again, and ask "Did you get heads on your second toss?"
- Regardless of whatever was asked, divide all answers into yes's and no's (with names written alongside each answer of course). Since a binary yes or no is all that is being recorded, the government, if suspicious, can easily contact them and verify they answered truthfully, without them actually knowing what it was they were being asked.
- Maths time!
Let's say we surveyed 1000 gazelles and got 450 yes's and 550 no's. The odds of a coin flip is 50:50, so give or take a margin of chance, 50% of the gazelles will not have been asked if they have the virus, and so got asked if they got heads on the second toss. And the odds of getting heads again is also 50:50, so those who got asked the other question should, give or take, return with 250 yes's and 250 no's.
We then deduct that from the main pile to get 200 yes's and 300 no's, to represent those who got asked if they have Geo-orgosis. But that's only half the surveyed population, so we double this, to 400 yes's and 600 no's.
As always, like any good survey, the larger your sample size, the more accurate and reliable your end results will be.
So we have our accurate survey of how far the virus spread, a record of everyone's answer, and nothing to suggest to the government that they have the virus.
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That is so damn simple. And we all missed it. Except for one smart muffin.
Okay, but how do we cure the disease?
*looks at the comments section*
Uhhhh...
You seem to have misunderstood me. I had a solution, but was led to believe, I needed 4 parts (3 breaks make 4 parts). Had it not been for the "three breaks" part, I would have given this as the solution: 1 link, 2-link, 4-link, 8-link, 16-link and 32-link (that actually was in my explanation and did work!). The reason I didn't is, that you had mentioned three breaks, and my solution would have proposed five.
But singling out one link from the chain is not a break, which happens between the links! You implied, that we need to single them out, but, had it not been for such a confusing definition, I wouldn't have proposed to use the 4-based system and would've gone with the 2-based one instead!
9422764
Oh...
I thought, it was four rooms... but, apparently, it is 4 by 4 = 16 rooms. I must have misunderstood the given
...
Welp. I do believe, my answer still counts as correct
... For is it not the same idea?
And this puzzle book should definitely find a place on one of my bookshelves
!
9508737
Why not? They won't answer "yes, we are ill" anyway, just that the chances of it are above 50%.
9508758
1. Because your solution makes the assumption there are some ponies who don't know if they are sick or not. Everyone being surveyed knows if they have Geo-orgosis, or not, for certain.
2. It does nothing to accurately establish the rate of the epidemic if you allow 'maybe' as an answer.
If there is no mention of it in the puzzle and nothing to infer as such, assume it is not meant to be part of the solution.
9508870
.
.
Yes, the surveyed know that for certain, but the king doesn't know about their illness for certain! Perhaps, if only those, who are not ill, answer, that no, they are not, it'll make more sense? That way the king would only know, that some gazelles are rather ill, than not, but that's not for sure. They don't answer "yes, I am ill", rather: "yes, I'm rather ill, than not (I'll just pretend I'm not sure)". So, if there are X ill gazelles (for sure!), there'll be a certain rate of x% of yesses, the other answers being noes. That would mean, x% of gazelles are ill (for sure!), but the king won't know (for sure) who, only that these x% are rather ill! So he won't banish any of them. And he'd assume, there's at the very least 51% chance of every "yes" being an ill gazelle, so that'd give (0,51*x)% accuracy, which is above half x. So at the very least more, than (x/2)%, are ill for sure
As you said yourself, it's the question, that makes the difference
9510265
You either have Geo-orgosis or you don't. There is no 'I have Geo-orgosis but only extra benign' option.
9510961
.
But they may not reveal they have! Therefore they may not answer "yes" to "Are you ill?". But they may (and must) still answer "yes" or "no", so be it another question, like the one I offered
9512117
Dude, there is NO question that fulfills the criteria of
1. Non-anonymous survey.
2. Accurately carried out survey.
3. Gets someone to outright admit having Geo-orgosis.
4. Contains no answers linking individuals to having Geo-orgosis.
If the answer is a 'maybe' and not even you can establish for certain if they have Geo-orgosis what's the point?
The other alternative question you posited is "Are the chances of you having Geo-orgosis above 50%?" Because there is only 2 binary choices (infected or not) then this might as well just be "Do you have the virus?" This single question cannot carry your survey. That's why I said in my answer the survey has to use TWO questions.
9512339
...
Well, your notion seems to be valid
But my question cannot come down just to "Have you or have you not?", because it leaves a possibility of having the virus, that's not just "1" or "0".
Then again, if two questions are required
... I wasn't supposed even to be solving this one, since I never passed Celestia's test from the previous chapter 
.