Here we go again. :-) ---------- Forwarded message --------- From: Alan Frank <alan@8wheels.org> Date: Sun, Nov 8, 2020 at 10:19 AM Subject: Brownie problem To: Jim Propp <JimPropp@gmail.com>, Jim Propp <jpropp@cs.uml.edu>, William Gasarch <gasarch@cs.umd.edu>, <laurenrose2@gmail.com> This e-mail originated from outside the UMass Lowell network. ---------------------------------------------------------------------- Math folks, On the one hand, I'd probably be in better shape if I didn't buy sweet pastries. On the other hand, they seem to lead to interesting problems. I was walking home yesterday, eating a brownie. They had given me an edge piece; I prefer inside pieces. I was wondering which there were more of; of course it depends on how the tray is cut. So here is a multi-part problem: 1. If exactly half the pieces are interior, what are the possible dimensions? 2. Same question if you extend the problem to three dimensions. 3. Continuing to higher dimensions, is there a closed-form expression for the number of possible solutions as a function of the dimension? I put the numbers that I knew into OEIS and did not see likely candidates. 4. Are there any interesting results if you vary the ratio? In two dimensions, there are six additional solutions for other integral ratios. --Alan