2. The squares of a $6 \times 6$ square grid are each labeled with a point value. As shown in the diagram below, the point value of the square in row $i$ and column $j$ is $i \times j$.
\begin{tabular}{|c|c|c|c|c|c|}
\hline 6 & 12 & 18 & 24 & 30 & 36 \\
\hline 5 & 10 & 15 & 20 & 25 & 30 \\
\hline 4 & 8 & 12 & 16 & 20 & 24 \\
\hline 3 & 6 & 9 & 12 & 15 & 18 \\
\hline 2 & 4 & 6 & 8 & 10 & 12 \\
\hline 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
\end{tabular}

A path in the grid is a sequence of squares, such that consecutive squares share an edge and no square occurs twice in the sequence. The score of a path is the sum of the point values of all squares in the path.
Determine the highest possible score of a path that begins with the bottom left corner of the grid and ends with the top right corner.
Solution: Let us colour the squares of the grid with a checkerboard pattern of black and white, starting with black in the bottom left-hand corner. A path in the grid will alternate between black and white squares, beginning and ending on black. Thus, any path cannot contain all 18 white squares. The path in the figure below contains all squares except for the white square with the lowest point value (2), so it will have the highest possible score.

The sum of the point values for all the squares is $(1+2+3+4+5+6)(1+2+3+4+5+6)=441$, and the score of the path is 439 .