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On Handkerchiefs
 > Egg Carton Theory
Positive and Negative

## Egg Carton Theory

Just in case there's not already enough evidence that I like to think about silly things, here's what I think about whenever I see the egg carton sitting in my refrigerator: what's the best way to arrange the eggs if I want to make the carton as much like a uniform solid as possible?

The question isn't as arbitrary as it sounds. The carton looks like a uniform solid, so I instinctively expect it to behave like one, too. I might set it on the edge of the counter with just the end sticking off, not realizing that all the weight is in that end; or I might pick it up loosely, get more torque than I expect, and fumble it. (I don't think I've ever dropped a whole carton of eggs, but I must have come close … why else would I have started thinking about this?) Anyway, the point is, I am motivated purely by practical concerns.

It wouldn't be hard to solve the problem analytically.

1. For each possible combination of eggs, work out the center of mass and moment of inertia (or perhaps the moments and axes of inertia).
2. Do the same for a uniform solid block.
3. Define the quantity you want to minimize … perhaps some differences squared, with weights.
4. See which combination of eggs is best.

But where's the fun in that? If I knew the answer, I wouldn't have anything to think about any more. Still, maybe I'll work it out some day when I'm bored.

Until then, instead of real answers, you get to see my current favorites.

Clearly the first egg has to be removed from the center. With a second egg removed, though, I much prefer to have them further out, as shown; sometimes I move the eggs around to achieve this. For three eggs, I'm not sure what to do. The combination in the picture is my best guess, but I still like to wonder about others. For example, you could move the egg in the upper left one space over! (The exclamation point indicates a good move, as in chess. Egg to king's rook four!) Usually, though, I settle for removing a center egg.

With four eggs, I really like the combination shown. It's probably not the best, but it's pretty good. I have no opinion about five eggs, so for that case I made a picture of what not to do. The six-egg combination is almost certainly the best possible. After that, I usually just take the eggs off the ends, even though it leads to the last two eggs being in the center.

I feel obliged to point out that in the first picture there are not eleven eggs, there are twelve eggs and one anti-egg.

As further evidence that I'm easily amused, here's the actual PostScript code for the first picture.

rcarton
0 1 egg 1 1 egg         3 1 egg 4 1 egg 5 1 egg
0 0 egg 1 0 egg 2 0 egg 3 0 egg 4 0 egg 5 0 egg
over

Finally, speaking of food, and of silly things I think about, it's always seemed to me that when you're making a sandwich, you ought to put the cheese on the mayonnaise side and the meat on the mustard side. I know it shouldn't make any difference to the taste, but there it is. I asked two people at work about it once, years ago … one completely agreed with me, the other thought I was out of my mind.