MathDB

Let

n

be a positive integer. In an

n\times4

table, each row is equal to
\begin{tabular}{| c | c | c | c |} \hline 2 & 0 & 1 & 0 \\ \hline \end{tabular}
A change is taking three consecutive boxes in the same row with different digits in them and changing the digits in these boxes as follows:

0\to1\text{, }1\to2\text{, }2\to0\text{.}

For example, a row \begin{tabular}{| c | c | c | c |}\hline 2 & 0 & 1 & 0 \\ \hline\end{tabular} can be changed to the row \begin{tabular}{| c | c | c | c |}\hline 0 & 1 & 2 & 0 \\ \hline\end{tabular} but not to \begin{tabular}{| c | c | c | c |}\hline 2 & 1 & 2 & 1 \\ \hline\end{tabular} because

0

1

, and

0

are not distinct.
Changes can be applied as often as wanted, even to items already changed. Show that for

n<12

, it is not possible to perform a finite number of changes so that the sum of the elements in each column is equal.

Problem Statement