TITLE: How minimally can I cut a cake sector-wise to fit it into a slightly undersized square tin?
QUESTION [12 upvotes]: In geometrical terms, what is the smallest number $n$ such that a disc of unit radius can be cut into $n$ sectors that can be reassembled without overlapping to fit into a square of side $2-\varepsilon$, where $\varepsilon>0$ is as small as you like? The question could be set for arbitrary straight cuts; but I like to serve my guests with traditional sector-shaped pieces.

REPLY [2 votes]: Here’s a third distinct three-piece configuration, with the centres of the sectors at the vertices of an equilateral triangle:
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All the pieces have an angle greater than $\frac{\pi}{2}$ (the angles in the diagram are $149.5^\circ$, $110^\circ$ and $100.5^\circ$).
A solution with three congruent pieces doesn’t seem possible.