An all-edge brownie pan: https://www.amazon.com/Bakers-Edge-Nonstick-Brownie-Pan/dp/B000MMK448/ On Sun, Nov 8, 2020 at 6:24 PM Marc LeBrun <mlb@well.com> wrote:
=Alan Frank They had given me an edge piece; I prefer inside pieces.
Nice problem! And this is an ideal collaboration for me, since I myself prefer to crunch the edgy bits!
If exactly half the pieces are interior, what are the possible dimensions? ...of course it depends on how the tray is cut.
Yes, so really this should be specified in more detail.
If we assume the tray is simply cut into unit squares, then the answer is 6-by-8, with a 4-by-6 block of 24 interior squares, surrounded by a 1 unit wide frame of 24 outside squares.
However if we relax this uniformity and allow other cuts then many new possibilities arise:
For example, you can divvy up an 8-by-8 as follows (pardon my poor ASCII art):
aabbbbcc aaAAAAcc dBCDEFGe dBCDEFGe dBCDEFGe dBCDEFGe ffHHHHhh ffgggghh
There are 8 interior pieces labeled A thru H and 8 exterior pieces (painted a thru h). Twelve pieces are 1-by-4s together with four 2-by-2s at the corners (painted a, c, f and h). Obviously each piece has area 4, and there are the same number inside as outside, so both the measure and the count are the same.
(Come to think of it, I like this better than the all-square division because this way all the outside pieces have 4 units of edge in contact with the baking pan, whereas the square way the corners get twice as crunchy.)
Puzzle: can you cut a tray so that ALL the brownies have the SAME non-square rectangular shape?
Extra credit: can you cut a tray so that all the brownies have the same area, but NONE of them have the same shape?
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