2007 APMO 第 5 题

A regular $(5 \times 5)$-array of lights is defective, so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state, from on to off, or from off to on. Initially all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all the possible positions of this light.

常规的 $(5 \times 5)$ 灯阵列是有缺陷的，因此切换一个灯的开关会导致同一行和同一列中的每个相邻灯以及灯本身改变状态，从开到关，或从关到开。最初所有灯都关闭。经过一定次数的切换后，只有一盏灯被打开。找出这盏灯所有可能的位置。