Maths/Stats challenge - 1 of 2 - polls

[EdS2]

Thanks Joe, that's some interesting material there, and thanks for linking it.

I think my case, where there are three or more poll responses, is a bit more involved than the case of just two responses. In effect, we have multiple constraints all of which need to be satisfied. I think this means we have more information and so we can get more confidence in the answers.

So, for example, when I found this example in the wild
A: 23%
B: 41%
C: 36%
the actual total number of votes was a single number, less than 50, and as it happens, none of the individual best approximations from each percentage (of 13, 17 and 11) are the right number. If we evaluate all three percentages for those three postulated number of votes, we get different results, perhaps for example:
A: 23% (3/13)
B: 38% (5/13)
C: 38% (5/13)
or
A: 24% (4/17)
B: 41% (7/17)
C: 35% (6/17)

So, I think there's more to it, to getting a best fit, or indeed the exact answer, in the case of polls with more than two candidates.

(Edit: I notice that the differences between results are a constraint too: in the case of 36% and 41%, the true difference must be at most 6%, and so we must have at least 16 votes.)