## The Three Light Bulbs Puzzle

There is a famous puzzle about three light bulbs, that is sometimes given at interviews.

Suppose that you are standing in a hallway next to three light switches that are all off. There is another room down the hall, where there are three incandescent light bulbs—each light bulb is operated by one of the switches in the hallway. You can’t see the light bulbs from the hallway. How would you figure out which switch operates which light bulb, if you can only go to the room with the light bulbs one and only one time?

This puzzle worked much better in the past when we only had incandescent light bulbs and so didn’t need to specify the type of bulbs. Unfortunately, the standard solution only works with incandescent bulbs and the word “incandescent” nowadays needs to be stated. But the use of “incandescent” is a big hint. Indeed, incandescent light bulbs generate heat when they are on, so the standard solution is to turn on the first light switch, to keep the second switch off, and to turn the third switch on for five minutes before turning it off. In the room, the light bulb corresponding to the first switch will be lit, and out of the two unlit bulbs, the one corresponding to the third switch will be warm.

It’s a cute solution, but there could be so many other approaches:

• You can ask a friend to help you. You can come to the room with the light bulbs and shout to her which switch to turn on.
• You can place mirrors from the room to the hall, so you can see bulbs through the multiple reflections. You might not need to enter the room at all.
• Before playing with the switches, you can place a video camera in the room to transmit the scene in real time. You will not be allowed to enter the room again to get the camera back, but what the heck: the job will be done.
• You can put timers on the switches and set them to turn on at different times.
• You can attach strings to switches and turn them on or off from a distance.

I invite my readers to invent other methods to solve this problem. Be creative. After all, if I were to interview you for a job, I would be more impressed by a new solution than the one that is all over the Internet.

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1. #### autismplusmath:

I saw this recently on the show “Brain Games” and had an “Oh, yeah!” moment when they showed someone touching the light bulbs to solve the problem.

I like the video camera approach since it turns on it’s head the assumption that you need to go to the light bulb room last, rather than first. I remember somewhere reading that you could rub certain fragrances on light bulbs in order to have the scent of lavender, vanilla, or whatever permeate the room. Perhaps I would rub light bulb #1 with the strong scent of cologne; I would wrap a tissue around light bulb #2; and I would remove light bulb #3 from its lamp. I would test each switch for a sufficiently long time to get a result: if I smelled cologne, the switch matched to light bulb #1; if I smelled smoke, or the sprinklers kicked in, the switch matched to light bulb #2; and if the room stayed dark, then the switch matched up to light bulb #3.

Thank you for the invitation–that was fun!

Glenn Laniewski
Blog:
autismplusmath
Related Post:
River Crossing Puzzle
https://autismplusmath.blogspot.com/2014/01/river-crossing-puzzle.html

2. #### Teo:

i would turn the first switch on and go down the hall to look at the bulbs. There I would unscrew one of the two unlit bulbs. Back at the switches I would turn the first switch back off and turn the second on. If no light came from the room I would know it was connected to the unscrewed bulb, and the last switch was connected to the remaining bulb.

3. #### George R.:

Hi Tanya!
Well known puzzle. One of few -i would say- that illustrates quite clearly the difference between Mathematics and real world (or “applied Mathematics,if you want..). I don’t see ANY way to solve it strictly mathematically (applying some kind of algorithm, permutation, or whatever):-)
Slight objection: I think there is no(commercial) source/device of light, that doesn’t produce at least a minimum amount of heat,when “On”. But, i’m not 100% sure about it.
Strong objection: Τhe same idea of having to attend an “interview” where such questions are going to be asked (trying to verify our “intelligence”..) is a social nightmare! 🙂

4. #### George R.:

Another solving proposal: Have a brand-new light bulb installed. Its average life expectancy is given by the producer (within some statistical approximation ,most probably following a normal (de Moivre-Gauss) distribution curve). Turn the switches on, with time intervals between the consecutive actions,greater or equal than the Variance of bulb life. Get enough food, clothing and a chemical toilet along with a laptop (in order to be able to visit Tanya’s blog!) and…the rest is trivial.
Proposal no.2: Turn all 3 on and go check! Υου have an 1/3 probability of being successful. Your chances of getting hired after this interview, are not higher,anyway… 🙂

5. #### George R.:

Ups..! I obviously meant “Turn either switch “ON” ” , at no.2 proposal. 🙂

6. #### George R.:

I withdraw my idiotic proposal 1. 🙂 , and submit an new one (not very original, since it is identical with an assumed one (urban legend) by Niels Bohr, but you’ve submitted already almost all the good ones, Tanya! )
One bribes (or flatters..) the housekeeper ,who definitely should know which #@%6\$*& bulb is the right one! If he/she doesn’t know, at least you will not be the only one who will be dismissed…

7. #### Fergal:

This immediately reminded me of the barometer problem. The story involves a physics student taking an exam which includes “Explain how to measure the height of a building using a barometer”. The student gives many clever answers, deliberately avoiding the one that is clearly the “right” answer. Good fun.

https://www.snopes.com/college/exam/barometer.asp

8. #### Pedro:

Turn the electricity off at the mains switchboard.
Label the switches A, B and C.
Turn switch A on; leave B and C off.
If all the switches are within a single fitting, disconnect one of the wires leading to switch B and connect it to either of the wires leading to switch A; if there are three separate switches use a spare piece of wire to make the same connection. (I’m assuming that we are dealing with a competent electrician with all the necessary gear for the job.)
Now go to the room where the lamps are.
For each of the three lamps there will be the pair of wires leading to the switch, a live and neutral in and live and neutral daisy-chained to another fitting. For each of the lamps in turn, use a continuity tester to check for a short circuit between between the wires leading to the switch; only one of the three lamps should have a short-circuit and this lamp is connected to switch A. Having establlished this, check for continuity between one of the wires leading to switch A and any of the other four switch wires; again there should onlky be one short-circuit and this wire leads to switch B. Obviously the remaining lamp is connected to switch C.
Go back to the switches (this is allowed; it’s only the lamps that have a limit of one visit) and undo the changes.

9. #### George R.:

In fact, it could be done practically perhaps even for 4 switches. We let 5 minutes or so for one switch and 0.5 minute for a second. The hardly warm “off”,the warmer”off”, the cold/off , the “on”. Theoretically , even for more switches, if there could be a “temperature-time” function of some kind (“exponential decay” sort of,perhaps?)

10. #### Animesh:

I use a stone/rock or a flying saucer…or a machinegun to knock of one off the bulbs. The switch which doesn’t turn off the light is switch 1…..
Then I break the other bulb…and same procedure
And I leave the third one…for sanity!

Someone already mentioned using a voltmeter(what I was gonna say) but did anyone consider just looking in the room without going in it? That seems simple, just open the door and don’t walk in… Someone mentioned just waiting till a light goes out, but then you’d still only have 2 states, on and off, and it would be just as confusing. Do they dim before going entirely out?

Besides that, you need a way to impart a unique state on exactly one light bulb that isn’t being either on or off. If it’s warm that works, but any other state is fine too, a weird smell, anything.

Another method that takes the order of flipping the switches and measuring the light bulbs out of the equation entirely: don’t use sight, smell or touch(and please don’t lick the light bulb either). By changing the voltage on the switch(assuming we are a competent, geared up electrician) the lights will either flicker differently, or by changing the bulbs first, can make a different flickering noise at the switch depending on the bulb used(it might be amps or watts depending on whether its an LED bulb or whatever else).

I suppose you could just rip the wiring out of the wall and follow it into the room too.

If you can see the light coming out the room when the bulbs are on, but not exactly where they are, and you can walk in and screw with stuff first, you can either tint the color(frequency) or just cover one bulb a bit and break the other(amplitude). The switch with no light corresponds to the broken bulb, some light the covered bulb, and the most intense of the 3 would be the untouched one. If you had a device to measure the light intensity precisely, and each bulb was a unique distance away, that would tell you which switch controls the furthers, closest, and middle bulbs. Any other properties unique to light? Polarization? I think that would be the same as a filter for frequencies(tinting) but with a polarizing filter.

Besides that, you’d need to move the light out of the room in a precise enough way to determine the location. Someone mentioned using reflection and mirrors, and I think the same would work in principle using refraction. An electronic sensor seems too straightforward.

Besides that if you can burn or ruin a bulb from the outside, that would add a state(on, off and ruined). How can you ruin it? Since there is not way to single out a bulb besides being on or off, turning it on would have to ruin just one somehow, and only once. Fill the room with water all the way to the ceiling(would require valves, pumps, and at least \$5,000 and an afternoon), light one bulb ruining it, drain the room, and light another. You now have 3 separate bulbs with 3 different states linked to 3 different switches. Or nothing happens, because the one you lit was screwed in tight enough to be watertight…

I thin that covers all the ways I know to measure things and screw with lights… In the strictest sense though, I can’t imagine any properties unique to light bulbs besides besides light and heat, and no way to manipulate those into more than 2 states from only inside the switch room(not counting the use of heat)…

12. #### Count Iblis:

You can use a method that is very similar to exploiting the heat produced by incandescent lightbulbs. Lightbulbs that are not incandescent will continue to radiate a very weak light due to fluorescence for a few minutes after being switched off. So, you can switch on one lightbulb for a few minutes while another one is left on. You go into the room, you then see which lightbulb in burning. You then unplug the other two lightbulbs and bring them to a dark room. Because your eyes won’t be dark adapted enough to see which one is glowing, you should take a picture of both the lightbulbs using a digital camera with high ISO setting and use a long exposure time (say 30 seconds at ISO of 800 or 1600, if you choose the ISO too high you’ll get too much noise).

With a digital camera, you should be able to detect extremely weak afterglow. Some pictures I took at ISO 800 with 60 seconds exposure time of the night sky shows stars of magnitude 11 or so, while you could perhaps spot stars up to magnitude 6 with the naked eye under the conditions the picture was taken. So, that’s a 100 times fainter than what you could see woth the anaked eye, while the afterglow is readily visible when you are dark adapted.

Solution that works for fluorescent but maybe not LED bulbs:

Turn the first switch on.

Flick the second switch on and off repeatedly, for at least as long as it takes to burn out most fluorescent bulbs; this might take a while, because they can last for far more duty cycles than an incandescent. Leave this switch on as well.

Leave the last switch alone (off).

Now, one bulb will be on (first switch), and two will be off. Unscrew an unlit bulb and stick your tongue in the socket. If you see a bright light at the end of a tunnel, this bulb is connected to the second switch…

14. #### Eddie:

You could still come to a conclusion regardless of what type of fixtures or bulbs are used. Since one light will be on when you walk into the room, it’s really between the other two. Having one of them on for 5-10 minutes may not create a lot of heat, but there will be a noticeable difference between the two remaining since one will be cold. Even 5 degrees difference can be felt.

15. #### Michael:

Yeah, what Eddie said. Turn the first 2 on for about 10 minutes, then turn the second off. then enter the room. The one on belongs to the first switch, the warm but dark one belongs to the second, and the cold belongs to the third.