Four, on a circular road, there are 4 middle schools arranged clockwise: $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$. They have 15, 8, 5, and 12 color TVs, respectively. To make the number of color TVs in each school the same, some schools are allowed to transfer color TVs to adjacent schools. How should the TVs be transferred to minimize the total number of TVs transferred? And what is the minimum total number of TVs transferred?
Let $A_{1}$ High School transfer $x_{1}$ color TVs to $A_{2}$ High School (if $x_{1}$ is negative, it means $A_{2}$ High School transfers $|x_{1}|$ color TVs to $A_{1}$ High School. The same applies below), $A_{2}$ High School transfer $x_{2}$ color TVs to $A_{3}$ High School, $A_{3}$ High School transfer $x_{3}$ color TVs to $A_{4}$ High School, and $A_{4}$ High School transfer $x_{4}$ color TVs to $A_{1}$ High School.
Since there are 40 color TVs in total, with an average of 10 per school, therefore,
$$
\begin{array}{l}
15-x_{1}+x_{4}=10,8-x_{2}+x_{1}=10, \\
5-x_{3}+x_{2}=10,12-x_{4}+x_{3}=10 . \\
\text { Then }=x_{1}-5, x_{1}=x_{2}+2, x_{2}=x_{3}+5, x_{3}= \\
x_{3}-2, x_{3}=\left(x_{1}-5\right)-2=x_{1}-7, x_{2}=\left(x_{1}-7\right)+5 \\
=x_{:}-2 .
\end{array}
$$

Then $x_{4}=x_{1}-5, x_{1}:=x_{2}+2, \tilde{x}_{2}=x_{3}+5, x_{3}=$ $=x:-\hat{2}$.
The problem requires $y=|x_{1}|+|x_{2}|+|x_{3}|+|x_{4}|$
$$
=|x_{1}|+|x_{1}-2|+|x_{1}-7|+|x_{1}
$$
$-5|$ to be minimized,
where $x_{1}$ is an integer satisfying $-8 \leqslant x_{1} \leqslant 15$.
Let $x_{1}=x$, and consider the function defined on $-8 \leqslant x \leqslant 15$
$$
y=|x|+|x-2|+|x-7|+|x-5| \text {, }
$$
$|x|+|x-7|$ represents the sum of the distances from $x$ to 0 and 7. When $0 \leqslant x \leqslant 7$, $|x|+|x-7|$ takes the minimum value 7; similarly, when $2 \leqslant x \leqslant 5$, $|x-2|+|x-5|$ takes the minimum value 3.

Therefore, when $2 \leqslant x \leqslant 5$, $y$ takes the minimum value 10. In other words, when $x_{1}=2,3,4,5$, $|x_{1}|+|x_{1}-2|+|x_{1}-7|+|x_{1}-5|$ takes the minimum value 10.

Thus, the minimum total number of color TVs transferred is 10, and there are four allocation schemes: