@Markiyan Hirnyk 

I know, and I answered to both versions. But you did not answer to my question.

@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!