You have two tasks. You must design a 3-switch lamp and a 4-switch lamp. It is recommended that you try the 3-switch lamp first. Your design will include a lightbulb, wires, switches and power sources. The design must follow these rules:

1. You may only use 1 lightbulb for each lamp. The 3-switch lamp can only have one power source, and the 4-switch lamp must have exactly two. You may use any number of wires.
2. Every flip of a switch, no matter the previous positions, must turn the lamp from on to off or off to on.
3. Each wire may connect to any number of switches, power sources, and other wires, and to the lightbulb.
4. Each switch has two separate positions to which wires can connect. If the switch is up, then all the wires connected to position 1 are considered connected to each other. If the switch is down, all the wires connected to position 2 are considered connected to each other.
5. The lightbulb turns on if and only if there exists a complete circuit that includes both the lightbulb and at least one power source.
6. A circuit is a sequence of wires, power sources, and the lightbulb where each is connected to the next item in the sequence (the last is connected to the first). No such sequence may list the same wire, power source, or the lightbulb twice.

I recommend that you denote the different wires with letters like A, B, C, etc.

First, I'm going to add one piece of hardware that is not in the problem as stated, but which makes a couple of things easier. You have an unlimited number of "junction points." It is easier to imagine three (or more) wires with one end attached to a junction point than to imagine a single three-ended (or more) wire. Cutting a wire and attaching the new ends to a junction point does not change the circuit. replacing a junction point with Position 1 of a switch does not change the circuit as long as the switch is in position 1.

Then I'm going to start with a simple circuit and add switches.

No switches: Wire A from + pole of battery to left connector of bulb; wire B from left connector of bulb to - pole of battery.

---- - Batt + -----(A)---- Bulb---
|                                 |
-------------------(B)------------

Add a switch: Cut wire B (which becomes wires B and C) and add junction point to restore circuit. Replace junction point with position 1 of switch.

---- (- Batt +) -------(A)---- (Bulb)---
|                                       |
|         P2                            |
-----(C)--P1-------------(B)------------

Add second switch: Cut wire C (which becomes wires C and D) and add junction point to restore circuit. Replace junction point with Position 1 of switch. This works as required except when both switches are in Position 2. Cut wires B and D and reconnect them with junction points 1 and 2. Add wire B' connecting junction point 1 to switch one at position 2, C' connecting Position 2 of each switch, and D' connecting switch 2 Position 2 to junction point 2.

  --(E)(- Batt +) -------(A)---- (Bulb)---
  |                                       |
(J2)--(D')--P2----(C')-----P2-----(B')---(J1)
  -----(D)--P1----(C)------P1-----(B)------

Adding a third switch by cutting wire E will give the full circuit utilizing position 1 of that switch.  Cutting wires F (the half of the "old" wire E connecting to the battery) and A we get junction points from which we can add wires duplicating the position 1 circuit for position 2, but then we have to worry about having added so many connections that we can't turn the light off from one of the wires. 

I suspect that starting with 4 switches it becomes impossible to just expand this way, and a more complex arrangement becomes necessary, which is why Tristan added a second battery.

Posted by TomM on 2005-05-10 07:31:16