If that's the case, then of course in 2 dimensions the best he can hope for is 3x3, which is 1/9 interior. There are only a few solutions. I guess there would be more solutions in higher dimensions. On Sun, Nov 8, 2020 at 5:00 PM James Propp <jamespropp@gmail.com> wrote:
More natural, yes, but too easy to find solutions. I think Alan's after the rarer situation in which the interior is small compared to the boundary. (I'll ask him.)
Jim
On Sun, Nov 8, 2020 at 4:45 PM Allan Wechsler <acwacw@gmail.com> wrote:
Wouldn't it be more natural to have 1/n of the pieces be on the border? For example, a 10x12 tray is 1/3 border, an 18x20 tray is 1/5 perimeter, an 18x32 tray is 1/6 border, an 18x56 tray is 1/7 perimeter...
On Sun, Nov 8, 2020 at 3:45 PM James Propp <jamespropp@gmail.com> wrote:
I'm pretty sure what Alan has in mind are situations in which 1/n of the pieces are interior, for some integer n.
Jim
On Sun, Nov 8, 2020 at 3:39 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
In Q4 --- "vary the ratio" of what, exactly?
(My culture gap might be showing here; at least it's not my waistline!) WFL
On 11/8/20, James Propp <jamespropp@gmail.com> wrote:
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.
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