Bounded Solution for Linear Diophantine Equation problem Given a set of linear Diophantine Equations (LDE's), where each equation is one of the following form:
Let C be a positive integer constant. Also, the number of variables in each equation is exactly $C$.


*

*$a_i + b_i+c_i.....=C$ or

*$a_i+b_i+c_i.....=(C+1)$


For every such set of LDE problem instance, the problem is solvable iff, at least one such solution exists, such that each variable's assigned value in that solution is:


*

*$\leq C$.

*$\geq 0$


In other words, the solution if it exists is bounded by the constant $C$ and $0$.
Can someone help with the proof of the above statement (or counterexamples with some small $C$)?
 A: So, if I understood properly your exposition, the system can be written as
$$
\left\{ \matrix{
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,c_{\,1} }  = c_{\,1}  + d_{\,1}  \hfill \cr 
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,c_{\,2} }  = c_{\,2}  + d_{\,2}  \hfill \cr 
  \quad \quad  \vdots  \hfill \cr 
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,c_{\,n} }  = c_{\,n}  + d_{\,n}  \hfill \cr}  \right.
$$
where $d_k \in \{0,1\}$.
Clearly we can always rearrange the rows by increasing $c_k$'s
$$
\left\{ \matrix{
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,c_{\,1} }  = c_{\,1}  + d_{\,1}  \hfill \cr 
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,c_{\,1} }  +  \cdots  + x_{\,c_{\,2} }  = c_{\,2}  + d_{\,2}  \hfill \cr 
  \quad \quad  \vdots  \hfill \cr 
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,c_{\,1} }  +  \cdots  + x_{\,c_{\,2} }  +  \cdots  +  \cdots  + x_{\,c_{\,n - 1} }  +  \cdots  + x_{\,c_{\,n} }  = c_{\,n}  + d_{\,n}  \hfill \cr}  \right.
$$
and subtracting from each row the preceding one
$$
\left\{ \matrix{
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,c_{\,1} }  = c_{\,1}  + d_{\,1}  \hfill \cr 
  x_{\,c_{\,1}  + 1}  +  \cdots  + x_{\,c_{\,2} }  = \left( {c_{\,2}  - c_{\,1} } \right) + d_{\,2}  - d_{\,1}  \hfill \cr 
  \quad \quad  \vdots  \hfill \cr 
  x_{\,c_{\,n - 1}  + 1}  +  \cdots  + x_{\,c_{\,n} }  = \left( {c_{\,n}  - c_{\,n - 1} } \right) + d_{\,n}  - d_{\,n - 1}  \hfill \cr}  \right.
$$
This is not, actually, a set of equations but a collection of $n$ independent equations,
since the variables in each row are different from (i.e. not related to) those in the other rows.
Now, an equation of those, of the type
$$
x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,m}  = m + \hat d\quad \left| {\;\hat d =  - 1,0,1} \right.
$$
can have solutions in which the various $x_k$ are not necessarily limited to the range $[0,m]$.
Thus your claim is not justified.
