This morning I had a waffle and a fried egg for breakfast. The fried egg was cooked with a mold, so it was perfectly round and three inches in diameter. The waffle was a 3-inch by 4-inch grid of one inch squares:
I ate the waffle one square at a time, cutting a piece of egg off with each piece. I did this fairly naïvely, centering the egg on the waffle and cutting along the edge of each square. This led to two pieces being entirely covered in egg, but the four corner pieces barely having any:
The next time I have this for breakfast, I want the egg more evenly divided between squares. I also don't want to have to get out a ruler to prepare breakfast! How can I divide the egg into 12 as-close-to-equal pieces meeting the following requirements:
- I can move the egg between cuts. The position of the egg must be defined by existing features such that it has no freedom to move. As a special case, the egg's rotation does not have to be constrained before the first cut.
- An edge of the egg can be constrained to be tangent to an edge of the waffle. This would still allow the egg to slide along the edge or rotate in place.
- A vertex of the waffle can be constrained to be coincident to an edge of the egg. The egg would still be free to rotate around the vertex or rotate in place.
- Once cut, a vertex of the egg can be constrained to be coincident to a vertex and/or edge of the waffle. The egg would only be free to rotate around that point and/or slide along that edge.
- Once cut, a straight edge of the egg can be constrained to be parallel to an edge of the waffle. It would be free to slide in two directions, but unable to rotate.
- All cuts must be straight lines, along an edge of a square. Cuts don't have to go through the entire egg, but must start and stop at corners of a square.
- The egg may be scored to keep pieces together and fully cut later.
- Each piece of egg must be a single contiguous piece.
- (Clarification) Don't worry about cutting the waffle - it should always end up as 12 squares. For the purposes of this puzzle, it only exists as a grid used to cut the egg.
Examples of constraints:
This egg is constrained by having two waffle-vertices coincident to its edge. While it is free to rotate, this doesn't matter as long as the egg is still whole:
This egg is also free to rotate, but is constrained by a point that it is coincident to, and an edge it is tangent to:
Once we start removing pieces of the egg, we also need to constrain rotation. This egg is NOT constrained:
However, if we add a rule saying these edges are parallel, it is:
Once the egg is cut, we can also use the new vertices as constraints. This egg is fully constrained by one waffle-vertex being coincident to an egg-vertex, and another waffle-vertex being coincident to an egg-edge:
I am not sure whether there is an exact solution to this puzzle. If an exact solution is found, it will be accepted. Otherwise, I will accept the answer with the lowest standard deviation between piece sizes on Friday, October 13 2023, which is World Egg Day.