@Markiyan Hirnyk 

You will have to alter a single line to obtain the other version of the problem.
But I suspect that you know this, so that I do not understand your intention with this problem. Was it a test for us? You should have mentioned this.

@Markiyan Hirnyk 

Why use DS for that?

Since you are interested only in the number of configurations, you may take any special case for the lines e.g. the displayed ones.

I wrote some "dirty" code and found that the number of configurations is 13 (or 9 if the points are not allowed in the same region).
I have not the patience to verify and clean the code, so I do not post it.

@Markiyan Hirnyk 

That is because the problem was not clear enough. If more than one point is permited in the same region, simply allow equal points in step 3 (so, 11^4 cases instead of binomial(11,4)).

@Markiyan Hirnyk 

The answer contains the algorithm. It is not difficult to program it in Maple.
Only the first step may need a little help from the simplex package if a complete automation is wanted.

@acer 

My point was: if in the Physics package `*` is redefined and the system is not affected, the why should we reject the same thing for `+`?

Of course, I also feel more comfortable if such basic operators are not redefined!

@acer 

Note also that the Physics package redefines `*`
So, as in your example, the conversion

convert([1,2,3], `*`);

will fail. Probably not a big loss.

There is a whole theory related to this sequence, see:

https://en.wikipedia.org/wiki/Logistic_map

@Mac Dude 

Yes, it would be nice to be ensured that a+b-b+c  is not going to be interpreted as a+c+c.

@Christopher2222 

I was told that the bug exists in Maple 11 (classic).

@Christopher2222 

Unfortunalely the bug exists in all versions.

The bug seems to be very old; I tested in Maple 14 (2010) and it is present!

@Carl Love 

Since f^(1/p) is needed only when f' = 0 ==> all the exponents are multiples of p, so their division by p is ok.

And  r = c^(1/p) is simple when p=3: r=0, 1 or 2; actually r=c (by Fermat).

It is not clear: how many equations do you have?
As I understand, there are n = i+1 unknowns: S[1],...,S[i+1].
Do you use the letter i  only for n-1? You should restate mathematically the problem.

@Axel Vogt 

The beauty of such solutions largely compensates the absence of a direct answer from Maple.
And that's why mathematicians must exist!

@Markiyan Hirnyk 

It is interesting to note that with OP's solve, Maple does itself the conversion to rationals but after that it enters some huge computations with lots of memory and Windows freezes.