Are you smarter than a 5th grader- Part 4

By splitting into cases,

$\color{#D61F06}{If}$ $\color{#D61F06}{you}$ $\color{#D61F06}{paint}$ $\color{#D61F06}{0}$ $\color{#D61F06}{boxes}$ , it can be done in 1 way.

$\color{#D61F06}{If}$ $\color{#D61F06}{you}$ $\color{#D61F06}{paint}$ $\color{#D61F06}{1}$ $\color{#D61F06}{box}$ , it also can be done in only 1 way.

$\color{#D61F06}{If}$ $\color{#D61F06}{you}$ $\color{#D61F06}{paint}$ $\color{#D61F06}{2}$ $\color{#D61F06}{boxes}$ , that has 2 ways to work with, 2 cases, if you paint adjacent ones then a way and if you paint diagonally opposite then one way - 2 ways

$\color{#D61F06}{If}$ $\color{#D61F06}{you}$ $\color{#D61F06}{paint}$ $\color{#D61F06}{3}$ $\color{#D61F06}{boxes}$ , it's same as not painting 1 box , same as case 2 hence 1 way.

$\color{#D61F06}{If}$ $\color{#D61F06}{you}$ $\color{#D61F06}{paint}$ $\color{#D61F06}{all}$ $\color{#D61F06}{4}$ $\color{#D61F06}{boxes}$ ,1 way.

Total $1+1+2+1+1 = \boxed{6}$

Looks like it is $4!/4$

This is because each coloring can be transformed in 4 ways by turning it $\frac{\pi}{2}$ radians everytime.