The three switches problem

The light bulb problem is a class of problem often presented at interviews as a test of lateral thinking. Here, in a mischievous spirit taken from the best subversive traditions we will attack this question by thinking a little too laterally …

There are different versions of the light bulb problem, but they rely on the same trick. Here is one version:

Your grand uncle lives in a rambling old house. He had an electrician over to wire a light in the attic but the electrician messed with the wiring and set up three switches instead of one. Your grand uncle wants to figure out which of the switches turns on the light in the attic. Problem is, you can’t tell from below whether the attic light is on or off. You can fiddle with the switches and then go up to the attic to inspect the light, but you can only do this once. The old coot says you can’t keep going back down and up the attic stairs fiddling with the light – apparently it disturbs the cat and he scratches the furniture.

The conventional answer to the question is to flip one switch on and leave it for a bit. Then turn it off and turn on another then go up to the attic. If the light is on, the last switch you flipped was the one. If the light is warm (Aha!) then the first switch you flipped is the one. If the light is off and cold, well is the remaining one. Let’s try and subvert the conventional answer to this problem by engaging in some mild subterfuge in the spirit of Calandra‘s “Angels-on-a-Pin“.

  1. Rapidly flip one of the switches on and off many times. Finally set this switch to the off position and flip another switch to on. Now go to the attic. If the light is on, it’s the currently on switch. If the light is burnt out, its the first switch.

  2. Call in your grand aunt, explain the problem to her, crawl up to the attic and have her flip the switches one by one. Yell when the light comes on.

  3. Flip a switch and inspect all the fixtures in the house, repeat and eliminate two of the switches. The third one is the attic switch. If none of the switches seem to be doing anything, where’s the problem? Just hit all three switches when you need the attic light on.

  4. Wait for a moonless night. Turn off all the lights in the house and draw all the curtains and blinds. Now flip the switches one by one. There will be a difference in the light level, however slight, from the attic light going on.

  5. Fly a drone up into the attic. Now you have eyes on target. Screw the cat.

  6. Take all bulbs out of their sockets and unplug all appliances. Now flip the switches one by one and watch the electric meter. This may take some patience.

  7. Open up the cover plate. Take a voltmeter and measure the drop in voltage as the switches are flipped. If only one has a voltage drop, that’s the one wired to the attic switch. If more than one have drops, throw stones into the attic until the light bulb is broken. Redo the measurements. The switch that now has no voltage drop (but did previously) is the one. Take a new bulb, go up into the attic and replace it.

  8. Flip each switch in turn and let the bulb warm up. Throw a bucket of water into the attic. If you hear a loud “POP” the bulb was on. Leave adequate cool down periods between tests. Take a new bulb, go up into the attic and replace it.

  9. Tie three strings to the switches. Go up into the attic with the strings. (Some will say this is subverting the question. Yes it is. Yes it is)

  10. Get all the shiny pots and pans your grand uncle has. Arrange them like mirrors so that you have a view of the attic from the switches.

As an aside, here is a problem I like better, as solving it actually teaches you some mathematical concepts:

Your grand uncle has passed on. In his will he’s bequeathed you an antique coin of immense value (so he says). He’s also left you with eight accurate forgeries of that coin that are indistinguishable from the original, except that they are just a fraction lighter. He’s also left you a balance, which can only be used twice before it breaks. So he says.

You are only allowed to take one of the coins with you. The original coin is priceless, the forgeries are worthless. So he says. You suspect all the coins are duds and the balance won’t break, but you take your Grand uncle’s will in the spirit it was meant, as a last fun puzzle to keep you busy so you don’t get too sad at the old codger’s passing.

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