GeekDad Puzzle of the Week — Eggsact Decartoning

This morning I continued our recent Sunday morning ritual: I made French Toast for myself and the kids. As I was getting the eggs out of the refrigerator, I quickly discovered that all of the eggs were at one end of the carton. Fortunately, all of the eggs were at the end I grabbed, so I just sort of whacked the bottom of shelf above instead of potentially dropping the whole carton of eggs.

So I started thinking… is there a “right” or a “best” way to remove eggs sequentially from a carton, so as to minimize this drop-force, or at least maximize the surprise of an unbalanced carton over time?

For the purposes of this puzzle, let’s assume that the amount of balance-force or drop-force for an egg in a given “dimple” is proportional to the square of the distance from the center of the dimple to the center of the carton. Let’s also assume that the each dimple is a square, that there is no non-dimple space in the carton, and that any given egg has circular sections so that it puts all of its balance-force in the center of the square. Additionally, let’s assume that force in any direction (either left-to-right or top-to-bottom) is equally bad, and that the carton is replace with a random end leading into the fridge; that is, we can’t count on either end being the one I grab, so for the purposes of this puzzle, assume that I always grab the carton by its center. Finally, as we go through a lot of eggs early imbalances have the same “weight” as late imbalances throughout the carton-clearing sequence.

At no point can the eggs in the carton be reshuffled. For this puzzle, you are removing eggs only.

Just looking at a carton of eggs, you can easily identify several “balanced” solutions. Using the diagram above for labels, the carton “AGFL” would be balanced for 4 eggs, as would “CJ.” Of course, any such solution is just one step in the overall process of clearing a carton.

This week’s GeekDad Puzzle of the Week is to determine the “best” sequence for emptying a single carton of eggs, one egg at a time, so as to minimize the amount of balance-force throughout the process. Answers will most likely look like a string of letters (i.e., ABCGHIDEFJKL is one way to empty a carton, but most likely not the most balanced) with one instance each of each letter A through L.

Please send your answers in to GeekDad Central for entry in the weekly drawing. Once again, the fine folks over at ThinkGeek are providing a \$50 Gift Certificate for our winner.

Good luck, and happy puzzling!