2. Every $\mathrm{cm}^{2}$ of a rectangle with dimensions $8 \mathrm{~cm}$ x $4 \mathrm{~cm}$ needs to be painted with a different color or shade. Is it possible to do this using red, blue, black, green, and yellow, as well as the shades that can be obtained by mixing two, three, four, or all five colors in equal proportions.
Solution: The rectangle has an area $\mathrm{P}=8 \times 4=32 \mathrm{~cm}^{2}$. With five colors, $5 \mathrm{~cm}^{2}$ can be painted. With two colors, 10 shades can be obtained, i.e., $10 \mathrm{~cm}^{2}$ can be painted. With three colors, 10 shades can also be obtained, i.e., another $10 \mathrm{~cm}^{2}$ can be painted. With four colors, 5 shades can be obtained, i.e., $5 \mathrm{~cm}^{2}$ can be painted. Finally, with all five colors, one shade can be obtained, i.e., $1 \mathrm{~cm}^{2}$ can be painted. Therefore, in total, $5+10+10+5+1=31 \mathrm{~cm}^{2}$ will be painted. This means $1 \mathrm{~cm}^{2}$ of the rectangle will remain unpainted with the given colors or shades obtained from them, taken in equal proportions?