How to solve this four variables system of equations?

Question

$$(x-1)^2+(y-1)^2+(z-1)^2=(t-1)^2,$$

$$(y+1)^2+(z+1)^2=(x+1)^2+(t+1)^2$$

Ηow to solve this for $(x,y,z,t)$ if $x<y<z<t$, $2\leq x, y, z, t\leq 16$ and if $x, y, z$ and $t$ are all natural numbers?

I don't how to start to solve this. Please help!

Answer 1

The problem could be solved in different ways:

The first one as mentioned by Gribouillis, could be doing an exhaustive search. In order to do so, different types of programming languages or software could be used. But you can also use an Access database and some Excel. The procedure will be as follows:

Create a table in Access, for example, name it Table1, and fill it with numbers from 2 to 16. So, one of the constraints will be achieved. Then use query designer to create the following query

SELECT    Table1.Field1
        , Table1_1.Field1
        , Table1_2.Field1
        , Table1_3.Field1
FROM  Table1
    , Table1 AS Table1_1
    , Table1 AS Table1_2
    , Table1 AS Table1_3
WHERE 
         Table1.Field1      <= Table1_1.Field1
    AND  Table1_1.Field1    <= Table1_2.Field1
    AND  Table1_2.Field1    <= Table1_3.Field1
;

The last part of the query, the WHERE section or conditions, makes sure that constraint $x<y<z<t$ holds. If done properly the resulting set would have 3060 records.

Finally use Excel to check every record and find, if any, hold both constraints

$$ (x-1)^2+(y-1)^2+(z-1)^2=(t-1)^2 \\ (y+1)^2+(z+1)^2=(x+1)^2+(t+1)^2 $$

And we arrived at the solution

x   y   z   t
3   7   10  12
3   6   15  16

A second way could be if we classified the problem as a Mixed Integer Nonlinear Programming. The Integer part comes from the constrain: "must be only natural numbers", the Nonlinear part of the problem is given by the constraints

$$ (x-1)^2+(y-1)^2+(z-1)^2=(t-1)^2 \\ (y+1)^2+(z+1)^2=(x+1)^2+(t+1)^2 $$

In order to solve it this way an objective function must be created, and a simple one could be

$$ MIN: x+y+z+t $$

So the optimization model would be complete and it would look like the following

$$ \begin{align} Min &: x+y+z+t \\ s.t. & \\ (x-1)^2+(y-1)^2+(z-1)^2&=(t-1)^2 \\ (y+1)^2+(z+1)^2&=(x+1)^2+(t+1)^2 \\ x&\leq y \\ y&\leq z \\ z&\leq t \\ x&\geq 2 \\ y&\geq 2 \\ z&\geq 2 \\ t&\geq 2 \\ x&\leq 16 \\ y&\leq 16 \\ z&\leq 16 \\ t&\leq 16 \\ \end{align} $$

$$ x,y,z,t \in \mathbb{N} $$

Then use some solver software like Premium Solver, What'sBest!, Lingo, etc.

Finally, as we've already classified the problem as MINLP several techniques could be used to solve it, it could be also possible to program a Genetic Algorithm to solve the problem.

I hope to give you some ideas

Answer 2

A line of research Exhaustive search show that the only sums of two squares that can be written in two different ways for distinct numbers in $[3, 17]$ are

130 {(3, 11), (7, 9)}
185 {(4, 13), (8, 11)}
205 {(6, 13), (3, 14)}
221 {(5, 14), (10, 11)}
250 {(9, 13), (5, 15)}
265 {(3, 16), (11, 12)}
305 {(7, 16), (4, 17)}
325 {(10, 15), (6, 17)}