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.

In the four switch problem, does the light bulb turn on iff AT LEAST one power source is part of a complete circuit involving the light bulb? Rule #5 would seem to indicate this.

Or, does the light bulb turn on iff EXACTLY TWO power sources are part of a single complete circuit involving the light bulb (as rule #1 may allow).

Or, does the light bulb turn on iff there are exactly two complete circuits that each involve exactly one power source and the light bulb?

If either of the two former, I am beginning to consider the hypothesis that there are no additional configurations of on and off that could be gained by having the additional power source. Every configuration of wires and powersources I can think of can be reproduced in some way with only a single power source. If I am correct both about this hypothesis and in my current interpretation of the rules of the problem (scenario #1 above), then if you have a solution to this problem that involves using both power sources, I suspect that it can be reduced to a single power source solution. However if the problem is actually the latter suggestion (the third interpretation proposed), then I can see how having a second power source will actually have a distinct effect, but I have been operating under the assumption that this is an incorrect interpretation of the rules.

Posted by Avin on 2006-01-23 09:28:14