equation solution in integer

[Bert Henry]

I have the equation

x + y = 15

an I'm looking for solution only in the range x=1..9 and y=1..9, x and y both integer

Is there a sage-command to do that?

Thanks in advance

Bert Henry

[Bert Henry]

I tried it with

var('x, y')

assume(x,"integer")

assume(x>0)

assume(y, "integer")

assume(y>0)

solve(x+y==15,x,y)

The result was

(t_0, -t_0 + 15)

obviously right, but not 6,9 7,8 8,7 and 9,6

[Matthias Koeppe]

http://doc.sagemath.org/html/en/reference/discrete_geometry/sage/geometry/polyhedron/base.html#sage.geometry.polyhedron.base.Polyhedron_base.integral_points

[Bert Henry]

@Matthias,

thanks for your answer, but I don‘t underrstand it. In your link, I can‘t fInd the solution for my problem. Would you give me a hunt, where to search?

Am Freitag, 17. April 2020 19:17:12 UTC+2 schrieb Bert Henry:

[slelievre]

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))

[Bert Henry]

wow, I didn‘t expect, that may „simple“ problem needs such deep math. I will look for the math of polyhedrons to understand, what you wrote, because in some number-crosswords (I don‘t know the correct english word) you search for solutions of the m entioned type. Also you need it in some amphanumerics like SEND+MORE=MONEY.

Thanks a lot for answering

Bert

Am Freitag, 17. April 2020 19:17:12 UTC+2 schrieb Bert Henry:

[Dima Pasechnik]

On Sun, Apr 19, 2020 at 7:41 AM Bert Henry <bert...@gmail.com> wrote:
>
>
> wow, I didn‘t expect, that may „simple“ problem needs such deep math. I will look for the math of polyhedrons to understand, what you wrote, because in some number-crosswords (I don‘t know the correct english word) you search for solutions of the m entioned type. Also you need it in some amphanumerics like SEND+MORE=MONEY.
>

my maths teacher pointed to us that when one compares numbers by
counting digits, one is actually doing logarithms in base 10 :-)

> Thanks a lot for answering
> Bert
>
>
> Am Freitag, 17. April 2020 19:17:12 UTC+2 schrieb Bert Henry:
>>
>> I have the equation
>> x + y = 15
>> an I'm looking for solution only in the range x=1..9 and y=1..9, x and y both integer
>> Is there a sage-command to do that?
>>
>> Thanks in advance
>> Bert Henry
>

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[Pedro A. Garcia]

May be you want to use ` WeightedIntegerVectors(15,[1,1])` or restricted partitions inside gap. This, for a single linear Diophantine equation like yours might be a fast approach.

Pedro

[Vincent Delecroix]

In this particular instance, I would rather write down the loop.
It is not hard to make the loop general for any bound on x and
y and any sum.

sum = 15
xmin = 1
xmax = 9
ymin = 1
ymax = 9
for x in range(max(xmin, sum-ymax), min(xmax, sum-ymin)+1):
print(x, sum-x)

which does run through

6 9
7 8
8 7
9 6

[Pedro A. Garcia]

Yes, @vdelecroix, in these cases this might be a very good option. 

sage: list( [x,y] for x in range(1,10) for y in range(1,10) if x+y==15)

[[6, 9], [7, 8], [8, 7], [9, 6]]

For a more general setting, normaliz, 4ti2 and other approaches are better; some are available in sage as pointed above.