# Dividing a connected shape into equal-sized connected pieces

It's nearly pancake day here in the UK and I'll be making my kids pancakes. The last one in the batch is always a little oddly shaped, and I'd like to divide it into two pieces of equal area so they can share it. However, the children are fussy eaters and insist on their pieces being path-connected. Is this possible? Maybe there's an old theorem somewhere that would help me have a stress-free day?

(n.b. I'm capable of making very precise and wiggly cuts with the tip of the knife, so model the cut as a continuous map $C:[0,1] \to \mathbb{R}^2$)

• Is one of them hairier than the other? Feb 17 '17 at 16:28
• I think for cutting in two, you can always find a straight cut, but I can't find the source at the moment. I'll post it as an answer when I find it. Feb 17 '17 at 16:36
• @SimonMarynissen. For many non-convex pancake shapes, you can’t find a straight cut that yields two path-connected pieces. worldartsme.com/images/spiral-background-clipart-1.jpg Feb 17 '17 at 16:56
• Yeah, I forgot the path-connected condition. Feb 17 '17 at 16:57

If you have $2^n$ kids, you can use a generalization of the Ham sandwhich theorem to cut it using only $n$ cuts. In $2$ dimensions this is: if you've $m$ pieces of pancakes, there exists a curve that splits all the $m$ pieces in two pieces of the same size (area). So if you got $2^n$ kids, you start by dividing that piece in $2$ and then those two pieces each in $2$ with one cut. You can repeat this to get $2^n$ pieces.