Question 36
Let $AB\Gamma$ be an equilateral triangle of side $k$ cm. We divide $AB\Gamma$ with parallel lines into $k^2$ small equilateral triangles of side $1$ cm. In this way, we create a grid (see figure for $k=7$). Inside every small triangle we put exactly one positive integer from $1$ to $k^2$, so that there are not two triangles with the same number. With vertices the points of the grid, we define regular hexagons of side $1$ cm. We call the value of a hexagon the sum of the numbers of its 6 small triangles. Find, as a function of $k$, the greatest possible value of the sum of the values of all hexagons.
