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Surely they can fit?
Find a heptagon with mirror symmetry that can tile a flat planeMax 4x1 pattern fit within 11x11 areaFit as many overlapping generators as possible
$begingroup$
Suppose you have a grid of squares that has even dimensions, with at least one dimension greater than or equal to 4 squares, and from one corner you remove a 1x4 rectangle of those squares
for example:
□□□□□□
□□□□□□
□□□□□□
XXXX□□
Can you fill in that grid using as many copies of the following shapes as you like?
(Each shape can be rotated any of the four ways, and can be flipped/mirrored)
□□
□□
□□
□□
□□□□
□□□□
If you can, provide an example solution. If you cannot, then you should provide a reasonable argument to why it can't be done.
tiling
asked 2 hours ago
micsthepickmicsthepick
2,49111127
$endgroup$
add a comment |
$begingroup$
Suppose you have a grid of squares that has even dimensions, with at least one dimension greater than or equal to 4 squares, and from one corner you remove a 1x4 rectangle of those squares
for example:
□□□□□□
□□□□□□
□□□□□□
XXXX□□
Can you fill in that grid using as many copies of the following shapes as you like?
(Each shape can be rotated any of the four ways, and can be flipped/mirrored)
□□
□□
□□
□□
□□□□
□□□□
If you can, provide an example solution. If you cannot, then you should provide a reasonable argument to why it can't be done.
tiling
asked 2 hours ago
micsthepickmicsthepick
2,49111127
$endgroup$
add a comment |
$begingroup$
Suppose you have a grid of squares that has even dimensions, with at least one dimension greater than or equal to 4 squares, and from one corner you remove a 1x4 rectangle of those squares
for example:
□□□□□□
□□□□□□
□□□□□□
XXXX□□
Can you fill in that grid using as many copies of the following shapes as you like?
(Each shape can be rotated any of the four ways, and can be flipped/mirrored)
□□
□□
□□
□□
□□□□
□□□□
If you can, provide an example solution. If you cannot, then you should provide a reasonable argument to why it can't be done.
tiling
asked 2 hours ago
micsthepickmicsthepick
2,49111127
$endgroup$
Suppose you have a grid of squares that has even dimensions, with at least one dimension greater than or equal to 4 squares, and from one corner you remove a 1x4 rectangle of those squares
for example:
□□□□□□
□□□□□□
□□□□□□
XXXX□□
Can you fill in that grid using as many copies of the following shapes as you like?
(Each shape can be rotated any of the four ways, and can be flipped/mirrored)
□□
□□
□□
□□
□□□□
□□□□
If you can, provide an example solution. If you cannot, then you should provide a reasonable argument to why it can't be done.
tiling
tiling
asked 2 hours ago
micsthepickmicsthepick
2,49111127
asked 2 hours ago
micsthepickmicsthepick
2,49111127
asked 2 hours ago
micsthepickmicsthepick
2,49111127
asked 2 hours ago
micsthepickmicsthepick
2,49111127
asked 2 hours ago
micsthepickmicsthepick
2,49111127
2,49111127
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, you cannot:
Color the grid like this.
Since the grid (before removal of the four cells) has even dimensions, it is made up of 2x2 blocks with each color once. So each color appears the same number of times.
All of the given shapes will always cover the same number of squares of each color: the first two cover one of each, and the last covers two of each. But after marking off the unused cells, the grid has more red and blue cells than yellow and green. So you can't cover the grid perfectly.
answered 43 mins ago
Deusovi♦Deusovi
64.1k6221277
$endgroup$
$begingroup$
that’s pretty much my reasoning!
$endgroup$
– micsthepick
30 mins ago
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
No, you cannot:
Color the grid like this.
Since the grid (before removal of the four cells) has even dimensions, it is made up of 2x2 blocks with each color once. So each color appears the same number of times.
All of the given shapes will always cover the same number of squares of each color: the first two cover one of each, and the last covers two of each. But after marking off the unused cells, the grid has more red and blue cells than yellow and green. So you can't cover the grid perfectly.
answered 43 mins ago
Deusovi♦Deusovi
64.1k6221277
$endgroup$
$begingroup$
that’s pretty much my reasoning!
$endgroup$
– micsthepick
30 mins ago
add a comment |
$begingroup$
No, you cannot:
Color the grid like this.
Since the grid (before removal of the four cells) has even dimensions, it is made up of 2x2 blocks with each color once. So each color appears the same number of times.
All of the given shapes will always cover the same number of squares of each color: the first two cover one of each, and the last covers two of each. But after marking off the unused cells, the grid has more red and blue cells than yellow and green. So you can't cover the grid perfectly.
answered 43 mins ago
Deusovi♦Deusovi
64.1k6221277
$endgroup$
$begingroup$
that’s pretty much my reasoning!
$endgroup$
– micsthepick
30 mins ago
add a comment |
$begingroup$
No, you cannot:
Color the grid like this.
Since the grid (before removal of the four cells) has even dimensions, it is made up of 2x2 blocks with each color once. So each color appears the same number of times.
All of the given shapes will always cover the same number of squares of each color: the first two cover one of each, and the last covers two of each. But after marking off the unused cells, the grid has more red and blue cells than yellow and green. So you can't cover the grid perfectly.
answered 43 mins ago
Deusovi♦Deusovi
64.1k6221277
$endgroup$
No, you cannot:
Color the grid like this.
Since the grid (before removal of the four cells) has even dimensions, it is made up of 2x2 blocks with each color once. So each color appears the same number of times.
All of the given shapes will always cover the same number of squares of each color: the first two cover one of each, and the last covers two of each. But after marking off the unused cells, the grid has more red and blue cells than yellow and green. So you can't cover the grid perfectly.
answered 43 mins ago
Deusovi♦Deusovi
64.1k6221277
answered 43 mins ago
Deusovi♦Deusovi
64.1k6221277
answered 43 mins ago
Deusovi♦Deusovi
64.1k6221277
answered 43 mins ago
Deusovi♦Deusovi
64.1k6221277
64.1k6221277
$begingroup$
that’s pretty much my reasoning!
$endgroup$
– micsthepick
30 mins ago
add a comment |
$begingroup$
that’s pretty much my reasoning!
$endgroup$
– micsthepick
30 mins ago
$begingroup$
that’s pretty much my reasoning!
$endgroup$
– micsthepick
30 mins ago
$begingroup$
that’s pretty much my reasoning!
$endgroup$
– micsthepick
30 mins ago
add a comment |
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