2001 Putnam B1

Let $n$ be an even positive integer. Write the numbers

$1,2,\ldots,n^2$ in the squares of an $n\times n$ grid so that the

$k$-th row, from left to right, is

$$(k-1)n+1,(k-1)n+2,\ldots, (k-1)n+n.$$

Color the squares of the grid so that half of the squares in each

row and in each column are red and the other half are black (a

checkerboard coloring is one possibility). Prove that for each

coloring, the sum of the numbers on the red squares is equal to

the sum of the numbers on the black squares.

令 $n$ 为偶正整数。写下数字

$1,2,\ldots,n^2$ 在 $n\times n$ 网格的方格中，这样

第 $k$ 行，从左到右，是

$$(k-1)n+1,(k-1)n+2,\ldots, (k-1)n+n。$$

给网格的方格涂上颜色，使每个方格中的一半都涂上颜色

行和每列都是红色，另一半是黑色(a

棋盘着色是一种可能性)。证明对于每个

着色，红色方块上的数字之和等于

黑色方块上的数字之和。