# “十字翻转棋问题” 的解法

## 题目描述

“传火” 这个游戏，需要点燃 $16$ 个火堆。

## 思路与题解

$$\begin{array}{|c|c|c|c|}\hline1&2&3&4\\\hline5&6&7&8\\\hline9&10&11&12\\\hline13&14&15&16\\\hline\end{array}$$

$$b=(1,1,1,0,1,0,0,1,1,0,0,1,0,1,1,1)^{\rm T}$$

$$b_1\oplus x_1\oplus x_2\oplus x_3\oplus x_4\oplus x_5\oplus x_9\oplus x_{13} = 0$$

$$x_1\oplus x_2\oplus x_3\oplus x_4\oplus x_5\oplus x_9\oplus x_{13} = b_1$$

$$\begin{cases}x_1\oplus x_2\oplus x_3\oplus x_4\oplus x_5\oplus x_9\oplus x_{13}&=&b_1\\x_1\oplus x_2\oplus x_3\oplus x_4\oplus x_6\oplus x_{10}\oplus x_{14}&=&b_2\\\qquad\qquad\cdots\\x_4\oplus x_8\oplus x_{12}\oplus x_{13}\oplus x_{14}\oplus x_{15}\oplus x_{16}&=&b_{16}\end{cases}$$

$$Ax=b\quad\Leftrightarrow\quad\left[\begin{array}{c}\textcolor{red}{1}\,\textcolor{red}{1}\,\textcolor{red}{1}\,\textcolor{red}{1}\,\textcolor{red}{1}\,0\,0\,0\,\textcolor{red}{1}\,0\,0\,0\,\textcolor{red}{1}\,0\,0\,0\\\textcolor{red}{1}\,\textcolor{red}{1}\,\textcolor{red}{1}\,\textcolor{red}{1}\,0\,\textcolor{red}{1}\,0\,0\,0\,\textcolor{red}{1}\,0\,0\,0\,\textcolor{red}{1}\,0\,0\\\cdots\;\text{another 13 lines}\;\cdots\\0\,0\,0\,\textcolor{red}{1}\,0\,0\,0\,\textcolor{red}{1}\,0\,0\,0\,\textcolor{red}{1}\,\textcolor{red}{1}\,\textcolor{red}{1}\,\textcolor{red}{1}\,\textcolor{red}{1}\end{array}\right]x=b$$