Several rectangles cover the unit square. Can I find a disjoint set of them whose area is at least $1/4$?












11














I am interested in the following question:




Let a finite sequence of rectangles in $mathbb{R}^2$ be given such that




  1. The edges of the rectangles are parallel to the coordinate axes, and


  2. The rectangles cover the unit square, $[0,1]^2$.



Is it possible to find, among these rectangles, a collection of mutually disjoint rectangles whose combined area is at least $1/4$?




As of yet, I'm not sure if a solution exists. My friend and I have spent a while thinking about this and have gotten nowhere.



Any help is greatly appreciated.










share|cite|improve this question






















  • Where'd you come across this problem? This is very nifty looking.
    – Steven Stadnicki
    Nov 24 at 2:52






  • 3




    This was asked before: math.stackexchange.com/questions/2381180/…
    – Alon Amit
    Nov 24 at 3:03






  • 1




    @AlonAmit - Close, but not quite: in that question, the ratio is to the total area covered by all the rectangles (thus the chosen solution, which uses overlapping rectangles covering areas of unbounded size). In this question, the ratio is to the unit square, so the solution there does not apply here.
    – Paul Sinclair
    Nov 24 at 15:23






  • 1




    @snulty Without loss of generality, we can assume that the rectangle is equal to the union, since we can just replace each rectangle by its intersection with the unit square.
    – Misha Lavrov
    Nov 28 at 3:46






  • 1




    @ChristianBlatter: It doesn't really matter... assume they're closed (so "cover the unit square" is as easy as possible) and that rectangles just sharing edges are considered disjoint (so "mutually disjoint" is as easy as possible), and the answer is still "no" (for any constant $c>0$, not just $1/4$).
    – mjqxxxx
    Nov 28 at 18:21
















11














I am interested in the following question:




Let a finite sequence of rectangles in $mathbb{R}^2$ be given such that




  1. The edges of the rectangles are parallel to the coordinate axes, and


  2. The rectangles cover the unit square, $[0,1]^2$.



Is it possible to find, among these rectangles, a collection of mutually disjoint rectangles whose combined area is at least $1/4$?




As of yet, I'm not sure if a solution exists. My friend and I have spent a while thinking about this and have gotten nowhere.



Any help is greatly appreciated.










share|cite|improve this question






















  • Where'd you come across this problem? This is very nifty looking.
    – Steven Stadnicki
    Nov 24 at 2:52






  • 3




    This was asked before: math.stackexchange.com/questions/2381180/…
    – Alon Amit
    Nov 24 at 3:03






  • 1




    @AlonAmit - Close, but not quite: in that question, the ratio is to the total area covered by all the rectangles (thus the chosen solution, which uses overlapping rectangles covering areas of unbounded size). In this question, the ratio is to the unit square, so the solution there does not apply here.
    – Paul Sinclair
    Nov 24 at 15:23






  • 1




    @snulty Without loss of generality, we can assume that the rectangle is equal to the union, since we can just replace each rectangle by its intersection with the unit square.
    – Misha Lavrov
    Nov 28 at 3:46






  • 1




    @ChristianBlatter: It doesn't really matter... assume they're closed (so "cover the unit square" is as easy as possible) and that rectangles just sharing edges are considered disjoint (so "mutually disjoint" is as easy as possible), and the answer is still "no" (for any constant $c>0$, not just $1/4$).
    – mjqxxxx
    Nov 28 at 18:21














11












11








11


7





I am interested in the following question:




Let a finite sequence of rectangles in $mathbb{R}^2$ be given such that




  1. The edges of the rectangles are parallel to the coordinate axes, and


  2. The rectangles cover the unit square, $[0,1]^2$.



Is it possible to find, among these rectangles, a collection of mutually disjoint rectangles whose combined area is at least $1/4$?




As of yet, I'm not sure if a solution exists. My friend and I have spent a while thinking about this and have gotten nowhere.



Any help is greatly appreciated.










share|cite|improve this question













I am interested in the following question:




Let a finite sequence of rectangles in $mathbb{R}^2$ be given such that




  1. The edges of the rectangles are parallel to the coordinate axes, and


  2. The rectangles cover the unit square, $[0,1]^2$.



Is it possible to find, among these rectangles, a collection of mutually disjoint rectangles whose combined area is at least $1/4$?




As of yet, I'm not sure if a solution exists. My friend and I have spent a while thinking about this and have gotten nowhere.



Any help is greatly appreciated.







combinatorics geometry combinatorial-geometry






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share|cite|improve this question











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asked Nov 24 at 2:45









Nathaniel B

786516




786516












  • Where'd you come across this problem? This is very nifty looking.
    – Steven Stadnicki
    Nov 24 at 2:52






  • 3




    This was asked before: math.stackexchange.com/questions/2381180/…
    – Alon Amit
    Nov 24 at 3:03






  • 1




    @AlonAmit - Close, but not quite: in that question, the ratio is to the total area covered by all the rectangles (thus the chosen solution, which uses overlapping rectangles covering areas of unbounded size). In this question, the ratio is to the unit square, so the solution there does not apply here.
    – Paul Sinclair
    Nov 24 at 15:23






  • 1




    @snulty Without loss of generality, we can assume that the rectangle is equal to the union, since we can just replace each rectangle by its intersection with the unit square.
    – Misha Lavrov
    Nov 28 at 3:46






  • 1




    @ChristianBlatter: It doesn't really matter... assume they're closed (so "cover the unit square" is as easy as possible) and that rectangles just sharing edges are considered disjoint (so "mutually disjoint" is as easy as possible), and the answer is still "no" (for any constant $c>0$, not just $1/4$).
    – mjqxxxx
    Nov 28 at 18:21


















  • Where'd you come across this problem? This is very nifty looking.
    – Steven Stadnicki
    Nov 24 at 2:52






  • 3




    This was asked before: math.stackexchange.com/questions/2381180/…
    – Alon Amit
    Nov 24 at 3:03






  • 1




    @AlonAmit - Close, but not quite: in that question, the ratio is to the total area covered by all the rectangles (thus the chosen solution, which uses overlapping rectangles covering areas of unbounded size). In this question, the ratio is to the unit square, so the solution there does not apply here.
    – Paul Sinclair
    Nov 24 at 15:23






  • 1




    @snulty Without loss of generality, we can assume that the rectangle is equal to the union, since we can just replace each rectangle by its intersection with the unit square.
    – Misha Lavrov
    Nov 28 at 3:46






  • 1




    @ChristianBlatter: It doesn't really matter... assume they're closed (so "cover the unit square" is as easy as possible) and that rectangles just sharing edges are considered disjoint (so "mutually disjoint" is as easy as possible), and the answer is still "no" (for any constant $c>0$, not just $1/4$).
    – mjqxxxx
    Nov 28 at 18:21
















Where'd you come across this problem? This is very nifty looking.
– Steven Stadnicki
Nov 24 at 2:52




Where'd you come across this problem? This is very nifty looking.
– Steven Stadnicki
Nov 24 at 2:52




3




3




This was asked before: math.stackexchange.com/questions/2381180/…
– Alon Amit
Nov 24 at 3:03




This was asked before: math.stackexchange.com/questions/2381180/…
– Alon Amit
Nov 24 at 3:03




1




1




@AlonAmit - Close, but not quite: in that question, the ratio is to the total area covered by all the rectangles (thus the chosen solution, which uses overlapping rectangles covering areas of unbounded size). In this question, the ratio is to the unit square, so the solution there does not apply here.
– Paul Sinclair
Nov 24 at 15:23




@AlonAmit - Close, but not quite: in that question, the ratio is to the total area covered by all the rectangles (thus the chosen solution, which uses overlapping rectangles covering areas of unbounded size). In this question, the ratio is to the unit square, so the solution there does not apply here.
– Paul Sinclair
Nov 24 at 15:23




1




1




@snulty Without loss of generality, we can assume that the rectangle is equal to the union, since we can just replace each rectangle by its intersection with the unit square.
– Misha Lavrov
Nov 28 at 3:46




@snulty Without loss of generality, we can assume that the rectangle is equal to the union, since we can just replace each rectangle by its intersection with the unit square.
– Misha Lavrov
Nov 28 at 3:46




1




1




@ChristianBlatter: It doesn't really matter... assume they're closed (so "cover the unit square" is as easy as possible) and that rectangles just sharing edges are considered disjoint (so "mutually disjoint" is as easy as possible), and the answer is still "no" (for any constant $c>0$, not just $1/4$).
– mjqxxxx
Nov 28 at 18:21




@ChristianBlatter: It doesn't really matter... assume they're closed (so "cover the unit square" is as easy as possible) and that rectangles just sharing edges are considered disjoint (so "mutually disjoint" is as easy as possible), and the answer is still "no" (for any constant $c>0$, not just $1/4$).
– mjqxxxx
Nov 28 at 18:21










1 Answer
1






active

oldest

votes


















10





+300









The question linked by @AlonAmit in the comments answers exactly this question, and shows that the answer (at least with the constant $1/4$) is no. For a concrete demonstration, start with a $6times 6$ square broken into thirty-sixths:
$$
begin{matrix}
0&1&2&3&4&5\
6&7&8&9&a&b\
c&d&e&f&g&h\
i&j&k&l&m&n\
o&p&q&r&s&t\
u&v&w&x&y&z
end{matrix}
$$

Now cover each corner $2times2$ by four individual $(1+varepsilon)times (1+varepsilon)$ rectangles, such that the four rectangles in the upper left ($0,1,6,7$) are mutually overlapping, as are those in each of the other corners. And cover the remaining shape in the center (a cross) by eight individual $(3+varepsilon)times(1+varepsilon)$ rectangles ($28e$, $39f$, $cde$, $ijk$, $kqw$, $lrx$, $fgh$, and $lmn$), such that all eight include the center of the square. Any disjoint set of these rectangles includes at most four of the $(1+varepsilon)times(1+varepsilon)$ rectangles and at most one of the $(3+varepsilon)times(1+varepsilon)$ rectangles, and so has total area just over $7/36approx 19.4%$ of the full square.






share|cite|improve this answer























  • Does a $(1+epsilon)times (1+epsilon)$ rectangle have area $1+2epsilon +epsilon^2$? Which is a big square?
    – snulty
    Nov 28 at 9:37










  • I would've thought theres arrangements where you cover 4 corners of the box, i.e. the four 2x2 squares, and the squares you've used are disjoint which would mean you've covered 16/36 of the area, since each letter represents 1/36 the area right? I must be misunderstanding something
    – snulty
    Nov 28 at 15:00












  • @snulty: By $(1+varepsilon)times(1+varepsilon)$, I mean covering a single small square plus a little extra. There's no disjoint set of the rectangles I listed that cover a corner $2times 2$, because each corner $2times 2$ is covered by four mutually overlapping rectangles, of which you can only include one.
    – mjqxxxx
    Nov 28 at 15:25










  • @DavidC.Ullrich: I changed it to say $6times 6$ from the outset.
    – mjqxxxx
    Nov 28 at 15:28






  • 5




    @mjqxxxx: I find this kind of answers very helpful, so I created an SVG image (nominal-animal.net/answers/unit-square-36.svg) you can use if you want to illustrate the covering. (Consider it public domain, or CC0-1.0 licensed.) The rectangles have rounded corners to make it easier to perceive the overlappings. The small squares areas are $frac{1+epsilon}{36} approx 0.0278$ of the unit square, and the larger rectangles $frac{3+epsilon}{36} approx 0.0834$. The largest disjoint sets contain four squares and one rectangle, for a maximum area of approx $0.1945$.
    – Nominal Animal
    Nov 29 at 3:56













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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









10





+300









The question linked by @AlonAmit in the comments answers exactly this question, and shows that the answer (at least with the constant $1/4$) is no. For a concrete demonstration, start with a $6times 6$ square broken into thirty-sixths:
$$
begin{matrix}
0&1&2&3&4&5\
6&7&8&9&a&b\
c&d&e&f&g&h\
i&j&k&l&m&n\
o&p&q&r&s&t\
u&v&w&x&y&z
end{matrix}
$$

Now cover each corner $2times2$ by four individual $(1+varepsilon)times (1+varepsilon)$ rectangles, such that the four rectangles in the upper left ($0,1,6,7$) are mutually overlapping, as are those in each of the other corners. And cover the remaining shape in the center (a cross) by eight individual $(3+varepsilon)times(1+varepsilon)$ rectangles ($28e$, $39f$, $cde$, $ijk$, $kqw$, $lrx$, $fgh$, and $lmn$), such that all eight include the center of the square. Any disjoint set of these rectangles includes at most four of the $(1+varepsilon)times(1+varepsilon)$ rectangles and at most one of the $(3+varepsilon)times(1+varepsilon)$ rectangles, and so has total area just over $7/36approx 19.4%$ of the full square.






share|cite|improve this answer























  • Does a $(1+epsilon)times (1+epsilon)$ rectangle have area $1+2epsilon +epsilon^2$? Which is a big square?
    – snulty
    Nov 28 at 9:37










  • I would've thought theres arrangements where you cover 4 corners of the box, i.e. the four 2x2 squares, and the squares you've used are disjoint which would mean you've covered 16/36 of the area, since each letter represents 1/36 the area right? I must be misunderstanding something
    – snulty
    Nov 28 at 15:00












  • @snulty: By $(1+varepsilon)times(1+varepsilon)$, I mean covering a single small square plus a little extra. There's no disjoint set of the rectangles I listed that cover a corner $2times 2$, because each corner $2times 2$ is covered by four mutually overlapping rectangles, of which you can only include one.
    – mjqxxxx
    Nov 28 at 15:25










  • @DavidC.Ullrich: I changed it to say $6times 6$ from the outset.
    – mjqxxxx
    Nov 28 at 15:28






  • 5




    @mjqxxxx: I find this kind of answers very helpful, so I created an SVG image (nominal-animal.net/answers/unit-square-36.svg) you can use if you want to illustrate the covering. (Consider it public domain, or CC0-1.0 licensed.) The rectangles have rounded corners to make it easier to perceive the overlappings. The small squares areas are $frac{1+epsilon}{36} approx 0.0278$ of the unit square, and the larger rectangles $frac{3+epsilon}{36} approx 0.0834$. The largest disjoint sets contain four squares and one rectangle, for a maximum area of approx $0.1945$.
    – Nominal Animal
    Nov 29 at 3:56


















10





+300









The question linked by @AlonAmit in the comments answers exactly this question, and shows that the answer (at least with the constant $1/4$) is no. For a concrete demonstration, start with a $6times 6$ square broken into thirty-sixths:
$$
begin{matrix}
0&1&2&3&4&5\
6&7&8&9&a&b\
c&d&e&f&g&h\
i&j&k&l&m&n\
o&p&q&r&s&t\
u&v&w&x&y&z
end{matrix}
$$

Now cover each corner $2times2$ by four individual $(1+varepsilon)times (1+varepsilon)$ rectangles, such that the four rectangles in the upper left ($0,1,6,7$) are mutually overlapping, as are those in each of the other corners. And cover the remaining shape in the center (a cross) by eight individual $(3+varepsilon)times(1+varepsilon)$ rectangles ($28e$, $39f$, $cde$, $ijk$, $kqw$, $lrx$, $fgh$, and $lmn$), such that all eight include the center of the square. Any disjoint set of these rectangles includes at most four of the $(1+varepsilon)times(1+varepsilon)$ rectangles and at most one of the $(3+varepsilon)times(1+varepsilon)$ rectangles, and so has total area just over $7/36approx 19.4%$ of the full square.






share|cite|improve this answer























  • Does a $(1+epsilon)times (1+epsilon)$ rectangle have area $1+2epsilon +epsilon^2$? Which is a big square?
    – snulty
    Nov 28 at 9:37










  • I would've thought theres arrangements where you cover 4 corners of the box, i.e. the four 2x2 squares, and the squares you've used are disjoint which would mean you've covered 16/36 of the area, since each letter represents 1/36 the area right? I must be misunderstanding something
    – snulty
    Nov 28 at 15:00












  • @snulty: By $(1+varepsilon)times(1+varepsilon)$, I mean covering a single small square plus a little extra. There's no disjoint set of the rectangles I listed that cover a corner $2times 2$, because each corner $2times 2$ is covered by four mutually overlapping rectangles, of which you can only include one.
    – mjqxxxx
    Nov 28 at 15:25










  • @DavidC.Ullrich: I changed it to say $6times 6$ from the outset.
    – mjqxxxx
    Nov 28 at 15:28






  • 5




    @mjqxxxx: I find this kind of answers very helpful, so I created an SVG image (nominal-animal.net/answers/unit-square-36.svg) you can use if you want to illustrate the covering. (Consider it public domain, or CC0-1.0 licensed.) The rectangles have rounded corners to make it easier to perceive the overlappings. The small squares areas are $frac{1+epsilon}{36} approx 0.0278$ of the unit square, and the larger rectangles $frac{3+epsilon}{36} approx 0.0834$. The largest disjoint sets contain four squares and one rectangle, for a maximum area of approx $0.1945$.
    – Nominal Animal
    Nov 29 at 3:56
















10





+300







10





+300



10




+300




The question linked by @AlonAmit in the comments answers exactly this question, and shows that the answer (at least with the constant $1/4$) is no. For a concrete demonstration, start with a $6times 6$ square broken into thirty-sixths:
$$
begin{matrix}
0&1&2&3&4&5\
6&7&8&9&a&b\
c&d&e&f&g&h\
i&j&k&l&m&n\
o&p&q&r&s&t\
u&v&w&x&y&z
end{matrix}
$$

Now cover each corner $2times2$ by four individual $(1+varepsilon)times (1+varepsilon)$ rectangles, such that the four rectangles in the upper left ($0,1,6,7$) are mutually overlapping, as are those in each of the other corners. And cover the remaining shape in the center (a cross) by eight individual $(3+varepsilon)times(1+varepsilon)$ rectangles ($28e$, $39f$, $cde$, $ijk$, $kqw$, $lrx$, $fgh$, and $lmn$), such that all eight include the center of the square. Any disjoint set of these rectangles includes at most four of the $(1+varepsilon)times(1+varepsilon)$ rectangles and at most one of the $(3+varepsilon)times(1+varepsilon)$ rectangles, and so has total area just over $7/36approx 19.4%$ of the full square.






share|cite|improve this answer














The question linked by @AlonAmit in the comments answers exactly this question, and shows that the answer (at least with the constant $1/4$) is no. For a concrete demonstration, start with a $6times 6$ square broken into thirty-sixths:
$$
begin{matrix}
0&1&2&3&4&5\
6&7&8&9&a&b\
c&d&e&f&g&h\
i&j&k&l&m&n\
o&p&q&r&s&t\
u&v&w&x&y&z
end{matrix}
$$

Now cover each corner $2times2$ by four individual $(1+varepsilon)times (1+varepsilon)$ rectangles, such that the four rectangles in the upper left ($0,1,6,7$) are mutually overlapping, as are those in each of the other corners. And cover the remaining shape in the center (a cross) by eight individual $(3+varepsilon)times(1+varepsilon)$ rectangles ($28e$, $39f$, $cde$, $ijk$, $kqw$, $lrx$, $fgh$, and $lmn$), such that all eight include the center of the square. Any disjoint set of these rectangles includes at most four of the $(1+varepsilon)times(1+varepsilon)$ rectangles and at most one of the $(3+varepsilon)times(1+varepsilon)$ rectangles, and so has total area just over $7/36approx 19.4%$ of the full square.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 28 at 15:26

























answered Nov 28 at 4:28









mjqxxxx

31k23985




31k23985












  • Does a $(1+epsilon)times (1+epsilon)$ rectangle have area $1+2epsilon +epsilon^2$? Which is a big square?
    – snulty
    Nov 28 at 9:37










  • I would've thought theres arrangements where you cover 4 corners of the box, i.e. the four 2x2 squares, and the squares you've used are disjoint which would mean you've covered 16/36 of the area, since each letter represents 1/36 the area right? I must be misunderstanding something
    – snulty
    Nov 28 at 15:00












  • @snulty: By $(1+varepsilon)times(1+varepsilon)$, I mean covering a single small square plus a little extra. There's no disjoint set of the rectangles I listed that cover a corner $2times 2$, because each corner $2times 2$ is covered by four mutually overlapping rectangles, of which you can only include one.
    – mjqxxxx
    Nov 28 at 15:25










  • @DavidC.Ullrich: I changed it to say $6times 6$ from the outset.
    – mjqxxxx
    Nov 28 at 15:28






  • 5




    @mjqxxxx: I find this kind of answers very helpful, so I created an SVG image (nominal-animal.net/answers/unit-square-36.svg) you can use if you want to illustrate the covering. (Consider it public domain, or CC0-1.0 licensed.) The rectangles have rounded corners to make it easier to perceive the overlappings. The small squares areas are $frac{1+epsilon}{36} approx 0.0278$ of the unit square, and the larger rectangles $frac{3+epsilon}{36} approx 0.0834$. The largest disjoint sets contain four squares and one rectangle, for a maximum area of approx $0.1945$.
    – Nominal Animal
    Nov 29 at 3:56




















  • Does a $(1+epsilon)times (1+epsilon)$ rectangle have area $1+2epsilon +epsilon^2$? Which is a big square?
    – snulty
    Nov 28 at 9:37










  • I would've thought theres arrangements where you cover 4 corners of the box, i.e. the four 2x2 squares, and the squares you've used are disjoint which would mean you've covered 16/36 of the area, since each letter represents 1/36 the area right? I must be misunderstanding something
    – snulty
    Nov 28 at 15:00












  • @snulty: By $(1+varepsilon)times(1+varepsilon)$, I mean covering a single small square plus a little extra. There's no disjoint set of the rectangles I listed that cover a corner $2times 2$, because each corner $2times 2$ is covered by four mutually overlapping rectangles, of which you can only include one.
    – mjqxxxx
    Nov 28 at 15:25










  • @DavidC.Ullrich: I changed it to say $6times 6$ from the outset.
    – mjqxxxx
    Nov 28 at 15:28






  • 5




    @mjqxxxx: I find this kind of answers very helpful, so I created an SVG image (nominal-animal.net/answers/unit-square-36.svg) you can use if you want to illustrate the covering. (Consider it public domain, or CC0-1.0 licensed.) The rectangles have rounded corners to make it easier to perceive the overlappings. The small squares areas are $frac{1+epsilon}{36} approx 0.0278$ of the unit square, and the larger rectangles $frac{3+epsilon}{36} approx 0.0834$. The largest disjoint sets contain four squares and one rectangle, for a maximum area of approx $0.1945$.
    – Nominal Animal
    Nov 29 at 3:56


















Does a $(1+epsilon)times (1+epsilon)$ rectangle have area $1+2epsilon +epsilon^2$? Which is a big square?
– snulty
Nov 28 at 9:37




Does a $(1+epsilon)times (1+epsilon)$ rectangle have area $1+2epsilon +epsilon^2$? Which is a big square?
– snulty
Nov 28 at 9:37












I would've thought theres arrangements where you cover 4 corners of the box, i.e. the four 2x2 squares, and the squares you've used are disjoint which would mean you've covered 16/36 of the area, since each letter represents 1/36 the area right? I must be misunderstanding something
– snulty
Nov 28 at 15:00






I would've thought theres arrangements where you cover 4 corners of the box, i.e. the four 2x2 squares, and the squares you've used are disjoint which would mean you've covered 16/36 of the area, since each letter represents 1/36 the area right? I must be misunderstanding something
– snulty
Nov 28 at 15:00














@snulty: By $(1+varepsilon)times(1+varepsilon)$, I mean covering a single small square plus a little extra. There's no disjoint set of the rectangles I listed that cover a corner $2times 2$, because each corner $2times 2$ is covered by four mutually overlapping rectangles, of which you can only include one.
– mjqxxxx
Nov 28 at 15:25




@snulty: By $(1+varepsilon)times(1+varepsilon)$, I mean covering a single small square plus a little extra. There's no disjoint set of the rectangles I listed that cover a corner $2times 2$, because each corner $2times 2$ is covered by four mutually overlapping rectangles, of which you can only include one.
– mjqxxxx
Nov 28 at 15:25












@DavidC.Ullrich: I changed it to say $6times 6$ from the outset.
– mjqxxxx
Nov 28 at 15:28




@DavidC.Ullrich: I changed it to say $6times 6$ from the outset.
– mjqxxxx
Nov 28 at 15:28




5




5




@mjqxxxx: I find this kind of answers very helpful, so I created an SVG image (nominal-animal.net/answers/unit-square-36.svg) you can use if you want to illustrate the covering. (Consider it public domain, or CC0-1.0 licensed.) The rectangles have rounded corners to make it easier to perceive the overlappings. The small squares areas are $frac{1+epsilon}{36} approx 0.0278$ of the unit square, and the larger rectangles $frac{3+epsilon}{36} approx 0.0834$. The largest disjoint sets contain four squares and one rectangle, for a maximum area of approx $0.1945$.
– Nominal Animal
Nov 29 at 3:56






@mjqxxxx: I find this kind of answers very helpful, so I created an SVG image (nominal-animal.net/answers/unit-square-36.svg) you can use if you want to illustrate the covering. (Consider it public domain, or CC0-1.0 licensed.) The rectangles have rounded corners to make it easier to perceive the overlappings. The small squares areas are $frac{1+epsilon}{36} approx 0.0278$ of the unit square, and the larger rectangles $frac{3+epsilon}{36} approx 0.0834$. The largest disjoint sets contain four squares and one rectangle, for a maximum area of approx $0.1945$.
– Nominal Animal
Nov 29 at 3:56




















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