Matthias is hinting at a possible reformulation
of the problem as finding integral points in a
polyhedron. Let me expand.
In RR^2, consider the set S of all (x, y) satisfying:
x >= 1
x <= 9
y >= 1
y <= 9
x + y = 15
or if one prefers,
-1 + x >= 0
9 - x >= 0
-1 + y >= 0
9 - y >= 0
-15 + x + y = 0
Since all the conditions used to define this set
are of one of the following forms:
(linear form in x and y) = 0
(linear form in x and y) >= 0
the subset S is what is called a "polyhedron" in R^2.
The problem in your original post can now be
rephrased as:
Find all integral points in the polyhedron S.
An introduction to polyhedra in Sage is at:
http://doc.sagemath.org/html/en/reference/discrete_geometry/sage/geometry/polyhedron/constructor.html
The polyhedron S can be input as
S = Polyhedron(ieqs=[[-1, 1, 0], [9, -1, 0], [-1, 0, 1], [9, 0, -1]],
eqns=[[-15, 1, 1]]),
Check that our input represents the correct polyhedron:
sage: print(S.Hrepresentation_str())
x0 + x1 == 15
-x0 >= -9
x0 >= 6
Find all integral points:
sage: S.integral_points()
((6, 9), (7, 8), (8, 7), (9, 6))
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