Using math for interior decorating with lamps When I was in college, I owned three lamps and had a dark apartment. I kept trying to position them in different areas of the room, but it was still dark. Then I decided to model the problem with math: given three light sources of equal strength in a rectangular solid of a room, where can you place them to maximize the average value of the amount of light reaching a point in the solid?
Then I realized that the asymptotes at the light bulbs themselves would make the average diverge.
Recently, I thought of a reformulation:
Given three light sources in a rectangular solid of a room (assume height of ten feet, width of 15 and length if 25),all at height 6 ft., where should they be placed to maximize the average value (Edit: Minimum value) of the intensity of the light on the plane at height 3.5 ft.? (Since the couches are that height). 
Alternatively, if someone can give a more accurate reformulation of this problem (perhaps involving reflections of the wall) and solve it, that would be much cooler.
 A: I realise this is an old question, but looking over the accepted answer I noticed a slight mistake; the minimum intensity in that room is actually in the top-left / top-right corners, but the model got stuck in the local minimum at the bottom (look at the contours). The actual solution, it turns out, is much more boring:

(that's with wall reflectivity of 0.4, which seems about standard for bright painted walls)
This was found by a dumb brute-force approach, dividing the room into discrete lamp positions and trying every possibility. That means no possibility of getting stuck in locally optimal solutions, but also means you have to trade speed for precision. You can find my implementation here (it's JavaScript):
http://htmlpreview.github.io/?https://github.com/davidje13/OptimisationStuff/blob/master/lights.htm
(for some reason it runs twice on that github.io viewer)
But perhaps the other solution is actually more appealing; the front wall is brighter, and the dark back corners aren't really important.
Well, after that I started messing about with the numbers, so here are some more findings:


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*4 lamps are best in a square:





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*5 lamps finally give something interesting: A squashed dice! (looks familiar as a speaker arrangement)





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*It looks like 6 lamps are best arranged in two lines of three, but I had to set the resolution quite low to calculate that, so I can't be certain)

*The darker the walls, the closer your lights should be to them (seems counter-intuitive to me), but the general pattern doesn't change.
Finally, for your original question the answer is actually quite simple: maximising the average intensity within the room is the same as maximising the total intensity. Since the bulbs are giving out a fixed amount of light no matter how you arrange them, and that light can only go to the ceiling, walls and floor, the problem becomes one of minimising the total light hitting the walls (ceiling and floor are roughly symmetric). This has an obvious solution: put all the lights in the middle of the room! Perhaps not very practical though. I think your reformulation is much more appropriate for getting balanced lighting.
