Changing Colors II

The applet below illustrates a problem of changing colors on a square board. This is an extension of the problem discussed elsewhere.

The applet displays a square array with one cell black. It permits changing colors simultaneously in any row, column, or a parallel to one of the diagonals. In particular, you can switch the color in a corner cell. (There is also an option to replace the black cells with -1 and all others with 1.)

Is it possible by a sequence of the allowed moves to obtain a monochrome board?

(In the applet, the arrows around the board are clickable.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

Discussion

References

A. Engel, Problem-Solving Strategies, Springer, 1998, p. 9

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

Check the "Show hint" box. Among the highlighted boxes, the moves allowed in the problem always change a pair of cells. I follows that if only one of them is black at the outset, their number will be odd after any sequence of moves.

With the ±1 incarnation, the product of numbers in the highlighted cells is invariant of the allowed moves; if at the beginning it was -1, it will remain -1 at any stage of there game.

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