Question 21
A $6\times6$ board has squares labeled by $(i,j)$. The source English describes $(1,1),\ldots,(1,6)$ as the first column, while the Chinese describes the first row; either convention gives the same conditions below. For each $k=0,1,\ldots,5$, the six squares $(i,j)$ with $i-j\equiv k\pmod6$ form one diagonal, so there are six diagonals. Can the numbers $1,2,\ldots,36$ be placed in the board so that:
(1) all column sums are equal;
(2) all row sums are equal;
(3) all diagonal sums are equal?
(1) all column sums are equal;
(2) all row sums are equal;
(3) all diagonal sums are equal?