Packing a cake into a box

TL;DR

This paper investigates the minimal number of cuts needed to fit a triangular cake with icing into a mirror-image box, proving that three cuts are sometimes necessary and providing examples for two-cut solutions.

Contribution

The paper confirms Boltyansky's conjecture that three cuts are sometimes required, and offers new examples of cakes that can be cut into two pieces.

Findings

Three cuts are sometimes necessary to fit the cake into the box.

Examples of cakes that can be cut into two pieces are provided.

The minimal number of cuts needed can be three, confirming Boltyansky's question.

Abstract

Given a cake in form of a triangle and a box that fits the mirror image of the cake, how to cut the cake into a minimal number of pieces so that it can be put into the box? The cake has an icing, so that we are not allowed to put it into the box upside down. V.G. Boltyansky asked this question in 1977 and showed that three pieces always suffice. In this paper we provide examples of cakes that cannot be cut into two pieces to put into the box. This shows that three is the answer to V.G. Boltyansky's question. Also we give examples of cakes which can be cut into two pieces.