15. Peter wants to colour the cells of a $3 \times 3$ square in such a way that each of the rows, each of the columns and both diagonals have cells of three different colours. What is the least number of colours Peter could use?
A 3
B 4
C 5
D 6
E 7
15. C Peter needs three different colours for the top row, say colours A, B, C. The central cell must be different from each of these (as it lies on the same diagonal as A and also of C, and in the same column as B), say colour D.
Suppose it is possible to use only these four colours. Note that the bottom left cell must be different from A (same column), and different from $\mathrm{C}$ and $\mathrm{D}$ (same diagonal), hence it must be colour B. But then the bottom right cell must be different from $\mathrm{A}$ and $\mathrm{D}$
\begin{tabular}{|c|c|c|}
\hline $\mathrm{A}$ & $\mathrm{B}$ & $\mathrm{C}$ \\
\hline & $\mathrm{D}$ & \\
\hline $\mathrm{B}$ & & $?$ \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|}
\hline A & B & C \\
\hline C & D & B \\
\hline B & C & E \\
\hline
\end{tabular}
(same diagonal), from $\mathrm{B}$ (same row) and from $\mathrm{C}$ (same column). Hence a fifth colour is needed. The arrangement above shows that five colours are sufficient.