Egg Carton Theory

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.

There are other questions I could have asked.

What's the best way to manage eggs? This is a fairly general question. To me, the main thing it suggests is that we could unload the eggs into another container. How about …

• An egg carton made of hard plastic, with no top and taller sides. My parents have one of these for some reason.
• An egg carton made of lead, so that the weight of the individual eggs is negligible.
• A combination egg carton and lazy Susan. If you don't need to pick the whole thing up, then it doesn't matter whether or not it's like a uniform solid.
• A ramp or tube, straight or helical, so that the eggs are emitted in “first in, first out” order (FIFO).

What's the best order to remove eggs in? This is the right question if you keep the eggs in the original carton and don't move them around. I tend to think about it in terms of finding the best path through the arrangements, maximizing the total score, or perhaps maximizing the minimum score. But, there are other strategies!