# Waffleing my Egg

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.

• Does the egg have to be cut while on top of the waffle? Can it be cut separately and then placed on the waffle? Can the waffle be cut separately from the egg? Commented Sep 20, 2023 at 18:35
• @GentlePurpleRain I don't think you can even place the egg not on top of the waffle, as valid egg positions must be constrained by waffle lines. You can't cut it not on top of the waffle, as all cuts have to be on waffle lines. I thought about the possibility of cutting the egg and waffle separately, but the waffle cuts actually seem like a red herring - the waffle is going to be cut into 12 squares in the end, and there is seemingly nothing that prohibits cutting along the same waffle edge twice, even if it's already cut. I think we only really need to worry about cutting the egg. Commented Sep 20, 2023 at 18:50
• @GentlePurpleRain Cuts have to be along waffle lines, so some part of the egg has to be over the waffle. You could, however, have part of the egg off the waffle, cut it along the edge of the waffle, and use the part that wasn't over the waffle somewhere else. Commented Sep 20, 2023 at 18:58
• @DqwertyC Can you explain further what it means to "constrain" the egg? Is the configuration shown in the picture a valid one to cut? From there, can you slide the egg to the left/right by an arbitrary amount? Or must the right/left edge align with a vertical waffle line? Can you align the right edge of the egg with the second vertical line and leave more than half the egg overhanging, or does that remove constraint in the vertical direction? Commented Sep 20, 2023 at 20:46
• I don't understand the first bullet point condition about moving the egg at all. It reads like waffle :) Could you please show example positions of the egg that are allowed, and those that are not? Commented Sep 20, 2023 at 21:05

Assuming I interpreted the rules correctly here is an exact solution:

It generates 30° pizza slices.

Only the creation of the first pizza slice is shown. The egg and incisions are blue except for the newest cut which is red. Alignment in step 2 uses making the first incision parallel to the horizontal grid. Step 4 aligns two corners (green circles) to the vertical grid and one to the horizontal grid. Panel 5 is not an actual step, it only highlights the result. Panel 6 shows an auxiliary right-angled triangle (teal) with side lengths $$2,1,\sqrt 3$$ from which it follows that its angles are 90°,30°,60°.

• This is really clever! If possible, could you add some measurements or trig to the final image to show that it is 30°? Commented Sep 22, 2023 at 16:49
• @DqwertyC I've added an explanatory panel. Commented Sep 22, 2023 at 17:28

We can get arbitrarily close to a perfect solution by making an arbitrarily large number of "tick marks" around the outer edge of the egg, and then cutting the egg into strips of the correct area.

To do this, start with the egg in its initial pictured configuration. Score the egg to make a "tick mark" at the 12 o'clock position. Now rotate the egg clockwise so that the tick mark is aligned with the next waffle line - the egg is constrained by the top and bottom points as well as the new tick mark anchor. The key to note is that the angle of rotation is irrational, since trigonometric functions of rational numbers are irrational, and we are finding the inverse cosine of 2/3. What this means is that the egg circumference is not evenly divisible by the rotation angle - no matter how many times we rotate the egg by this angle, we'll never arrive back at the same rotation angle. We can keep rotating the egg and making tick marks at 12 o'clock, and will get an ever-higher density of notches around the egg to whatever level we desire.

EDIT: I'm no longer sure this proves it's possible, since the circumference of a circle is also irrational. We need to show that the angle doesn't evenly divide $$\pi$$ to be able to get an infinite density of tick marks, irrationality of the angle alone won't do it.

Once that is done, we have an arbitrary number of anchor points around the edge of the egg, so we can basically align it on the waffle however we want. Now we just need to cut the egg into strips so that each strip has area arbitrarily close to $$2.25\pi/12$$, which is easily done since we can draw any arbitrary chord across the egg and have it be a valid cut.