Consider an undecorated cylindrical cake and a perfect knife. We want to find the center of the cross-sectional circle.
If we can only score the surface of the cake, this reduces to finding the center of a circle with just a straightedge, which is impossible.
But using a knife allows additional constructions. For example, Sarvesh Iyer mentions:
The big difference between straight lines on a circle and knives on a cake is that you can shift any cake pieces you cut, around the shape and match them up with other pieces. The piece that's in a darker shade of orange can be removed and used to replicate that particular angle around the center subtended by it. I don't think one can create duplicate angles with just a straightedge, hence the difference.
Similarly, YNK claims that
determines the center in seven cuts, although he has not explained the construction procedure.
Here are some rules modeling our use of the knife.
- You can't guarantee any nice properties of the lines like being perpendicular or parallel to another, just chords.
- The "canvas" for connections is only the circle (you can't be cutting the table, just the cake).
What is the minimal number of cuts necessary to find the circle? Is it YNK's 7?
Best Answer
The seven figures given above show how we can determine the center of a cake by drawing seven line segments with a knife. In a few hours time, I will add text to this answer describing things like why we need to cut a portion segment that subtend an angle greater than $90^o$. So the interested parties can get themselves ready with a cake and knife similar to those shown in the diagrams given below.