Numbers in boxes (but not Sudoku)
I’ve been taking a random walk through Engel’s Problem Solving Strategies– reading it linearly is more of a chore than fun. The best reading method seems to be, read the first couple paragraphs of each chapter where the principle is introduced, and skip to the problems; you can look back at the worked examples later if you get stuck on applying the principle.
The first chapter deals with using invariants as a problem solving technique. This technique seems to be useful for dealing with problems which ask whether a specified state can be reached by following an algorithm. My favorite problem so far (of those I’ve managed to solve) that employs this principle is:
You may switch the signs of all numbers of a row, column, or a parallel to one of the diagonals. In particular, you may change the sign of each corner square. Prove that at least one -1 will remain in the table.
![\left[
\begin{matrix}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & -1 & 1 & 1
\end{matrix}
\right]](/cz/latexrender/pictures/4a1bcce3ea8766bd055126b08d9ef5ec.png "\left[
\begin{matrix}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & -1 & 1 & 1
\end{matrix}
\right]")
Possibly relevant posts:
- Linear Recursion Equations (11/24/2005)
- problems out of context (3/17/2007)
- Full-rank underdetermined systems (11/2/2006)