Allan, Alan confirms that he means that 1/n of the pieces are in the interior for some positive integer n. So, in the 2D case, we’re looking at the equation ab=n(a-2)(b-2) or (a/(a-2))(b/(b-2))=n. If we look at the deceasing sequence of fractions 3/1, 4/2, 5/3, 6/4, 7/5, ... and note that (7/5)(7/5) < 2, we see that the only way two of such fractions can multiply to a whole number is if at least one of them belongs to {3/1, 4/2, 5/3, 6/4}. The solutions I get are (3/1)(3/1)=9 (3/1)(4/2)=6 (3/1)(5/3)=5 (3/1)(8/6)=4 (4/2)(4/2)=4 (4/2)(6/4)=3 (5/3)(12/10)=2 (6/4)(8/6)=2 Did I miss any? (The fact that there are six different numbers that appear to the right of an equals sign in my table may be what Alan meant by “six additional solutions for other integral ratios”, but I’d be inclined to think of this as eight solutions, or maybe fourteen if we “un-mod-out” by symmetry.) Jim On Sun, Nov 8, 2020 at 4:59 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.
--Alan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun