1992 CMO 第 3 题

Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an *adjustment* as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite *adjustments*.

给定 $9\times 9$ 网格，我们将 $+1$ 或 $-1$ 分配给网格上的每个方格。我们定义一个*调整*如下：对于网格上的每个正方形，我们将与该正方形共享公共边(不包括其本身)的所有数字进行乘积。然后我们有 $81$ 乘积。接下来，我们将所有方块的值替换为其相应的乘积。确定是否可以通过有限的*调整*使网格中的所有值都等于$1$。