Fix $1\leq d\in\mathbb{N}$ and set $D:=\{0,1,\ldots,d-1\}$. Consider the system of equations \begin{equation} x_i=c_i + \sum_{j\in D}\delta_{x_j,i} \end{equation} with $c_i\in D$ given and $x_i\in D$ unknown for $i\in D$. $\delta_{i,j}=\left\{\begin{array}{ll}1, & i=j \\ 0, & i\not= j\end{array}\right.$ denotes the Kronecker symbol.

## Question.

Characterize those $c=(c_i)_{i\in D}$ for which the above equation has a (unique) solution $x=(x_i)_{i\in D}$.

One can obviously use computers to answer this question for small $d$. This is not what I am after. The question might very well be hard. In that case I am also interested in partial results that cover the case $d=10$.

## Background.

One can show that \begin{equation} \sum_{i\in D} x_i =\sum_{i\in D} c_i + d \end{equation} and \begin{equation} \sum_{i\in D} i x_i =\sum_{i\in D} i c_i+\sum_{i\in D} c_i + d. \end{equation} These constraints suffice to show that there are no solutions for $d=2$ and that in case $d=3$ the only 4 possible solutions are \begin{equation} c=\left(\begin{array}{c}1\\0\\0\end{array}\right),x=\left(\begin{array}{c}1\\2\\1\end{array}\right) \mbox{ or } x=\left(\begin{array}{c}2\\0\\2\end{array}\right) \end{equation} or \begin{equation} c=\left(\begin{array}{c}1\\1\\0\end{array}\right),x=\left(\begin{array}{c}1\\2\\2\end{array}\right) \end{equation} or \begin{equation} c=\left(\begin{array}{c}2\\0\\0\end{array}\right),x=\left(\begin{array}{c}2\\1\\2\end{array}\right). \end{equation} The above constraints can also be use to put some mild constraints on $c$ and $x$ like \begin{equation} \sum_{i\in D}c_i\leq d(d-2) \end{equation} and \begin{equation} x_i \leq c_i+ \frac{\sum_{j\in D} c_j + d}{i} \end{equation} for $0<i\in D$. To answer the question one probably needs more sophisticated tools from additive number theory or combinatorics.

## Motivation.

The motivation for such equations originally stems from the compilation of self-referential forms (cf. the bounty question https://math.stackexchange.com/questions/464868/does-there-exist-a-general-solution-of-this-counting-numbers-game/484663#484663). The above remarks imply that you can compile the form

Date: 0.

This form contains ___ zeros.

This form contains ___ ones.

This form contains ___ twos.

as

Date: 0.

This form contains 1 zero.

This form contains 2 ones.

This form contains 1 two.

or

Date: 0.

This form contains 2 zeros.

This form contains 0 ones.

This form contains 2 twos.

However, compiling

Date: 2.

This form contains ___ zero.

This form contains ___ ones.

This form contains ___ two.

is not possible.