$\begingroup$
I want to cut a hexagon paper into regions of equal areas (more precisely either into squares of side c or into regular hexagons of side c). In both cases some of the papers will be wasted. Is it possible to know what is the best way to waste the minimum of papers? (Maybe something related to the Honeycomb conjecture?)
$\endgroup$ 22 Answers
$\begingroup$The answer will very much depend on whether squares or hexagons, and how many. For example, if you specify $7$ squares in a regular hexagon, then these two arrangements are possible:
The partition on the right is superior (squares are larger; wastes less paper), but I don't know if it is optimal.
Update. Here is an improved $7$-square packing, as suggested by Aaron Meyerowitz:
$\endgroup$ 1 $\begingroup$
Here are a few nontrivial examples of hexagons in hexagons:
