[math-fun] Fwd: Brownie problem
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
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
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
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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
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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
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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.
--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
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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
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=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?
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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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
Nice problem! This is close to a problem I worked on many years ago. Consider Alan’s question 2 (3 dimensions). What integer dimensioned bricks (rectangular parallelepipeds) have surface area equal to volume? Not too hard, so extend: what pairs of integer bricks have the surface area of one equal to the volume of the other, and vice versa? Surface-volume pairs: kind of a cousin to the brownie problem. I don’t have the list at hand, but there are only a handful, maybe a dozen give or take a factor of 2. A pair of 6x6x6 cubes is an obvious degenerate case. It’s a puzzling Diophantine situation, with 2 equations in 6 variables. If you vary one parameter, possible solutions form a series of points more and more closely spaced on the number line, sort of like a “chirp”. If you vary a different parameter, you get a chirp going the other way on the number line. Solutions are where points in both chirps coincide. So you can evaluate the points in one chirp as they get more closely spaced, until some break-even point where it’s better to evaluate the possible solutions based on the other chirp. This is the only time I’ve used floating point math with error bounds in the program to decide whether the floating point was within acceptable precision. Sorry this description is vague, it was long ago. — Mike
On Nov 8, 2020, at 6:56 PM, James Propp <jamespropp@gmail.com> wrote:
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
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participants (6)
-
Allan Wechsler -
Fred Lunnon -
James Propp -
Marc LeBrun -
Mike Beeler -
Mike Stay