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.

So, although I find this problem very interesting, I haven't made any significant progress on it in the past few days. Here's what I've been working with.

In order to have the light turn on at the appropriate times, the light must be ON for 1+,2+,3+, OFF for any of the three permutations where one switch is - and the other two are +, ON for the three permutations where one switch is + and the other two are -, and OFF for 1-,2-,3-.  (Or the reverse of this, where the light's status is reversed, but the two situations are identical)

This means that there must be a minimum of four circuits, one for 1+,2+,3+, one for 1+,2-,3-, one for 1-,2+,3-, one for 1-,2-,3+.

As for the way current flows through these circuits, I am guessing they are of the form:

PS > 1+ > 2+ > 3+ > LB > 3+ > 2+ > 1+ > PS
PS > 1- > 2+ > 3- > LB > 3- > 1- > 2+ > PS
PS > 1+ > 2- > 3- > LB > 3- > 1+ > 2- > PS
PS > 1- > 2- > 3+ > LB > 3+ > 2- > 1- > PS

I.e., the circuit requires passing through each switch on the way to the light bulb and back. However the above does not work because if we had the minimum wires to make it work, with the switches in the 1-,2-,3- position, we can make a circuit as follows:

PS > 1- > 2- > 3- > LB > 3- > 1- > PS

Thus turning the lightbulb on, when it should be off.

I am considering writing a computer program to try to solve this, even though I normally don't like doing so, just because this puzzle has me intrigued enough!

Posted by Avin on 2005-05-16 13:58:08