# Challenging Problem

• November 22nd 2009, 10:59 AM
amm345
Challenging Problem
An ingenious engineer constructed the following switchboard: It is a rectangular array of 3×3 switches each of which can be ON or OFF, but the wiring is such that if a button is pressed all neighbouring buttons toggle their state as well. Thus, for instance if the the centre button is pressed, also the buttons at its west, north, south and east positions are pressed; if the top right button is pressed, this affects also the top centre, and right centre buttons (in matrix notation, the positions (1, 2) and (2, 3)).
a) Is it possible, beginning with the state of all switches ON, by pushing some buttons to arrive at the state of all switches OFF?

b) Is it possible, beginning with any state, to arrive at the state of all switches OFF?
• November 22nd 2009, 11:15 AM
qmech
Does this work?

$
A=\left(\begin{array}{ccc}1&1&1\\1&1&1\\1&1&1\\\en d{array}\right)
$

Click the (1,1) entry
$
A=\left(\begin{array}{ccc}0&0&1\\0&1&1\\1&1&1\\\en d{array}\right)
$

Click the (1,3) entry
$
A=\left(\begin{array}{ccc}0&1&0\\0&1&0\\1&1&1\\\en d{array}\right)
$

Click the (3,3) entry
$
A=\left(\begin{array}{ccc}0&1&0\\0&1&1\\1&0&0\\\en d{array}\right)
$

Click the (3,1) entry
$
A=\left(\begin{array}{ccc}0&1&0\\1&1&1\\0&1&0\\\en d{array}\right)
$

Click the (2,2) entry
$
A=\left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\\\en d{array}\right)
$
• November 22nd 2009, 11:18 AM