Question about solutions to simultaneous equation

[simonarcher99]

Would nayone be able to give me an answer and an explanation to this question. I have realised if the 9 variables are arranged in a 3 by 3 grid, each row and column has to add to five. Thankyou

How many distinct solutions consisting of positive integers does this system of equations have?
a+b+c=5
d+e+f=5
g+h+i=5
a+d+g=5
b+e+h=5
c+f+i=5

 

[Deleted member 4993]

Would nayone be able to give me an answer and an explanation to this question. I have realised if the 9 variables are arranged in a 3 by 3 grid, each row and column has to add to five. Thankyou

How many distinct solutions consisting of positive integers does this system of equations have?
a+b+c=5
d+e+f=5
g+h+i=5
a+d+g=5
b+e+h=5
c+f+i=5

Are the numbers unique (i.e. NOT equal to each other)?

 

[Cartesius24]

Look out

This equation is on positive integers. So we serarch on set of \(\displaystyle \{ 1,2,3,4 \} \)

 

[simonarcher99]

Are the numbers unique (i.e. NOT equal to each other)?

The numbers can be equal to each other just how many different solutions are there with positive integers where 0 is not a positive integer.

 

[simonarcher99]

a = b = c = d = e = f = g = h = i = 3/5

This'll work (in alpha order):
1,1,3,1,2,2,3,2,0

A unique solution requires additional info...like a+c = 2e

I don't think 0 can be used as it isn't a positive integer. A solution for example would be 1,2,2,2,1,2,2,2,1. But how many solutions are there?

 

[soroban]

How many distinct solutions consisting of positive integers does this system of equations have?

\(\displaystyle \begin{array}{cvcc}a+b+c\:=\:5 && a+d+g\:=\:5 \\
d+e+f\:=\:5 && b+e+h \:=\:5 \\
g+h+i\:=\:5 && c+f+i \:=\:5 \end{array}\)

I found twelve.

\(\displaystyle \begin{pmatrix}a&b&c \\ d&e&f \\ g&h&i \end{pmatrix}\;=\;
\begin{pmatrix}1&2&2\\2&1&2 \\ 2&2&1\end{pmatrix} \quad \begin{pmatrix}1&2&2\\2&2&1\\2&1&2\end{pmatrix} \quad \begin{pmatrix}2&1&2\\1&2&2\\2&2&1\end{pmatrix} \quad \begin{pmatrix}2&1&2\\2&2&1\\1&2&2\end{pmatrix} \quad
\begin{pmatrix}2&2&1\\1&2&2\\2&1&2\end{pmatrix} \quad \begin{pmatrix}2&2&1\\2&1&2\\1&2&2\end{pmatrix}\)

. . . . . . . . . . . \(\displaystyle =\;\begin{pmatrix}1&1&3\\1&3&1\\3&1&1\end{pmatrix} \quad \begin{pmatrix}1&1&3\\3&1&1\\1&3&1 \end{pmatrix} \quad \begin{pmatrix}1&3&1\\1&1&3\\3&1&1\end{pmatrix} \quad \begin{pmatrix}1&3&1\\3&1&1\\1&1&3\end{pmatrix} \quad \begin{pmatrix}3&1&1\\1&1&3\\1&3&1 \end{pmatrix} \quad \begin{pmatrix}3&1&1\\1&3&1\\1&1&3 \end{pmatrix} \)

 

[pka]

Would nayone be able to give me an answer and an explanation to this question. I have realized if the 9 variables are arranged in a 3 by 3 grid, each row and column has to add to five.

ys
I found twelve.
\(\displaystyle \begin{pmatrix}a&b&c \\ d&e&f \\ g&h&i \end{pmatrix}\;=\;
\begin{pmatrix}1&2&2\\2&1&2 \\ 2&2&1\end{pmatrix} \quad \begin{pmatrix}1&2&2\\2&2&1\\2&1&2\end{pmatrix} \quad \begin{pmatrix}2&1&2\\1&2&2\\2&2&1\end{pmatrix} \quad \begin{pmatrix}2&1&2\\2&2&1\\1&2&2\end{pmatrix} \quad
\begin{pmatrix}2&2&1\\1&2&2\\2&1&2\end{pmatrix} \quad \begin{pmatrix}2&2&1\\2&1&2\\1&2&2\end{pmatrix}\)

Look at the part in blue. These are \(\displaystyle 3\times 3\) matrices, entries positive integers less than four, each column and each row add to five. Soroban has given a nice list. Note the each matrix solution, \(\displaystyle \mathcal{M}\), its transpose \(\displaystyle \mathcal{M}^T\) is also a solution. There each of these is made from set \(\displaystyle \{3,1,1\}\) OR \(\displaystyle \{1,2,2\}\). Because of the requirement that rows & columns must each add to five, the odd-element-out must appear exactly once in any row or any column. How many \(\displaystyle 3\times 3\) matrices with entries \(\displaystyle 3\text{ or }1\) where \(\displaystyle 3\) must appear exactly once in any row or any column?

Dennis does that answer your question?

 

[Ishuda]

Would nayone be able to give me an answer and an explanation to this question. I have realised if the 9 variables are arranged in a 3 by 3 grid, each row and column has to add to five. Thankyou

How many distinct solutions consisting of positive integers does this system of equations have?
a+b+c=5
d+e+f=5
g+h+i=5
a+d+g=5
b+e+h=5
c+f+i=5

Should simple rotations be counted as the same solution?