Wire n switches, a power source and a lamp so that switching any one of the switches changes the state of...

physics/n.switches New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Wire n switches, a power source and a lamp so that switching any one of
the switches changes the state of the lamp (if the lamp is ON it will go
OFF, no matter which switch you use).

All the switches have only one inlet and one outlet and they have two
states (ON/OFF, or UP/DOWN, whatever you decide to call them).  No
multiple in- or outlets are allowed in the power source nor in the
lamp.  You may, however, use as much wire as you need and make as many
_permanent_ connections between the wires as you need (that is, one
wire may branch into several other wires, or vice versa).

What is the solution for n=3, if any?  Can it be done in plane, or does
it require other kinds of surfaces, or perhaps more dimension than two?

-- mkivihar@lk-hp-6.hut.fi

Show solution

Wire up some DPDT (Double Pole Double Throw) switches to act as
bypass-exchance networks.  The switch would have two positions:

In 1   -------------   Out 1

In 2   -------------   Out 2

            or

In 1   -----\ /-----   Out 1
             X
In 2   -----/ \-----   Out 2

Wire however many switches up in series.  Then wire power to one rail
at one end, and the light bulb to any rail at the other end (run a long
wire back to the power supply).  Then, any switch in the network can
change the on/off polarity.

P.S.  A DPDT switch looks like this:

                  0------0      0
   Position 1
                  0------0      0


                  0      0------0
   Position 2
                  0      0------0

So a bypass/exchange switch can be wired the following way:

              In 1
                |
Out 1 --0       0      0-- Out 2

Out 2 --0       0      0-- Out 1
                |
              In 2

-- Henry Houh <hhh@dill.lcs.mit.edu>

[ Editor: a DPDT switch has more than the one inlet and one outlet the
problem allows; no posting in the collection answers the problem under
that restriction. ]