How many colors does it take?
$begingroup$
This question is from a popular monthly science magazine in my country:
You have an 8x8 square where any 3 squares forming a tromino (including reflections and rotations) must consist of three different colored squares. At least how many colors do you need to paint the entire board?
Bonus: How about larger boards?
graph-theory polyomino
edited Feb 11 at 11:11
Bass
31k472188
asked Feb 11 at 9:54
NautilusNautilus
4,078525
$endgroup$
add a comment |
$begingroup$
This question is from a popular monthly science magazine in my country:
You have an 8x8 square where any 3 squares forming a tromino (including reflections and rotations) must consist of three different colored squares. At least how many colors do you need to paint the entire board?
Bonus: How about larger boards?
graph-theory polyomino
edited Feb 11 at 11:11
Bass
31k472188
asked Feb 11 at 9:54
NautilusNautilus
4,078525
$endgroup$
$begingroup$
Re: the edits, I was sure that the right name was triomino (based on tri(o)-), but I had put the word "popular" right before "science" because I wanted to stress that the magazine deals with popular science, so I don't think it necessarily breaks the adjective order rule.
$endgroup$
– Nautilus
Feb 11 at 14:41
add a comment |
$begingroup$
This question is from a popular monthly science magazine in my country:
You have an 8x8 square where any 3 squares forming a tromino (including reflections and rotations) must consist of three different colored squares. At least how many colors do you need to paint the entire board?
Bonus: How about larger boards?
graph-theory polyomino
edited Feb 11 at 11:11
Bass
31k472188
asked Feb 11 at 9:54
NautilusNautilus
4,078525
$endgroup$
This question is from a popular monthly science magazine in my country:
You have an 8x8 square where any 3 squares forming a tromino (including reflections and rotations) must consist of three different colored squares. At least how many colors do you need to paint the entire board?
Bonus: How about larger boards?
graph-theory polyomino
graph-theory polyomino
edited Feb 11 at 11:11
Bass
31k472188
asked Feb 11 at 9:54
NautilusNautilus
4,078525
edited Feb 11 at 11:11
Bass
31k472188
asked Feb 11 at 9:54
NautilusNautilus
4,078525
edited Feb 11 at 11:11
Bass
31k472188
edited Feb 11 at 11:11
Bass
31k472188
edited Feb 11 at 11:11
Bass
31k472188
31k472188
asked Feb 11 at 9:54
NautilusNautilus
4,078525
asked Feb 11 at 9:54
NautilusNautilus
4,078525
asked Feb 11 at 9:54
NautilusNautilus
4,078525
4,078525
$begingroup$
Re: the edits, I was sure that the right name was triomino (based on tri(o)-), but I had put the word "popular" right before "science" because I wanted to stress that the magazine deals with popular science, so I don't think it necessarily breaks the adjective order rule.
$endgroup$
– Nautilus
Feb 11 at 14:41
add a comment |
$begingroup$
Re: the edits, I was sure that the right name was triomino (based on tri(o)-), but I had put the word "popular" right before "science" because I wanted to stress that the magazine deals with popular science, so I don't think it necessarily breaks the adjective order rule.
$endgroup$
– Nautilus
Feb 11 at 14:41
$begingroup$
Re: the edits, I was sure that the right name was triomino (based on tri(o)-), but I had put the word "popular" right before "science" because I wanted to stress that the magazine deals with popular science, so I don't think it necessarily breaks the adjective order rule.
$endgroup$
– Nautilus
Feb 11 at 14:41
$begingroup$
Re: the edits, I was sure that the right name was triomino (based on tri(o)-), but I had put the word "popular" right before "science" because I wanted to stress that the magazine deals with popular science, so I don't think it necessarily breaks the adjective order rule.
$endgroup$
– Nautilus
Feb 11 at 14:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I think the answer is
5
using the following coloring:
For other board sizes,
5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)
Reasoning:
Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a tromino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.
answered Feb 11 at 10:02
GlorfindelGlorfindel
14.4k45486
$endgroup$
$begingroup$
I think it's wrong, the angular triomino "2 5 3" (with the 5 at the angle) is repeating. The 2 on first line, 2nd column, and the 2 on 4th line 3rd column, they make the same triomino with the same colours.
$endgroup$
– Rémi Henry
Feb 11 at 10:09
$begingroup$
They are all repeating (since it's a pattern); the point is that each triomino has three different colours. Triominos are not compared with each other (at least that's how I interpreted the puzzle).
$endgroup$
– Glorfindel
Feb 11 at 10:11
1
$begingroup$
@Oray what's wrong with just extending the pattern?
$endgroup$
– Glorfindel
Feb 11 at 10:39
1
$begingroup$
If at all possible, it's good to use something else than different shades of red and green for conveying significant information.
$endgroup$
– Bass
Feb 11 at 11:23
1
$begingroup$
Yeah, good idea.
$endgroup$
– Glorfindel
Feb 11 at 11:31
|
show 7 more comments
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I think the answer is
5
using the following coloring:
For other board sizes,
5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)
Reasoning:
Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a tromino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.
answered Feb 11 at 10:02
GlorfindelGlorfindel
14.4k45486
$endgroup$
$begingroup$
I think it's wrong, the angular triomino "2 5 3" (with the 5 at the angle) is repeating. The 2 on first line, 2nd column, and the 2 on 4th line 3rd column, they make the same triomino with the same colours.
$endgroup$
– Rémi Henry
Feb 11 at 10:09
$begingroup$
They are all repeating (since it's a pattern); the point is that each triomino has three different colours. Triominos are not compared with each other (at least that's how I interpreted the puzzle).
$endgroup$
– Glorfindel
Feb 11 at 10:11
1
$begingroup$
@Oray what's wrong with just extending the pattern?
$endgroup$
– Glorfindel
Feb 11 at 10:39
1
$begingroup$
If at all possible, it's good to use something else than different shades of red and green for conveying significant information.
$endgroup$
– Bass
Feb 11 at 11:23
1
$begingroup$
Yeah, good idea.
$endgroup$
– Glorfindel
Feb 11 at 11:31
|
show 7 more comments
$begingroup$
I think the answer is
5
using the following coloring:
For other board sizes,
5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)
Reasoning:
Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a tromino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.
answered Feb 11 at 10:02
GlorfindelGlorfindel
14.4k45486
$endgroup$
$begingroup$
I think it's wrong, the angular triomino "2 5 3" (with the 5 at the angle) is repeating. The 2 on first line, 2nd column, and the 2 on 4th line 3rd column, they make the same triomino with the same colours.
$endgroup$
– Rémi Henry
Feb 11 at 10:09
$begingroup$
They are all repeating (since it's a pattern); the point is that each triomino has three different colours. Triominos are not compared with each other (at least that's how I interpreted the puzzle).
$endgroup$
– Glorfindel
Feb 11 at 10:11
1
$begingroup$
@Oray what's wrong with just extending the pattern?
$endgroup$
– Glorfindel
Feb 11 at 10:39
1
$begingroup$
If at all possible, it's good to use something else than different shades of red and green for conveying significant information.
$endgroup$
– Bass
Feb 11 at 11:23
1
$begingroup$
Yeah, good idea.
$endgroup$
– Glorfindel
Feb 11 at 11:31
|
show 7 more comments
$begingroup$
I think the answer is
5
using the following coloring:
For other board sizes,
5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)
Reasoning:
Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a tromino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.
answered Feb 11 at 10:02
GlorfindelGlorfindel
14.4k45486
$endgroup$
I think the answer is
5
using the following coloring:
For other board sizes,
5 is sufficient as well; the pattern can just be repeated. (Of course, a 2x2 board needs only 4 colors because there are only 4 squares. And does 1x1 even count as a board?)
Reasoning:
Consider a square not on the edge of the board with its 4 orthogonal neighbours; they all have to have different colors since each pair is part of a tromino. Therefore, we need at least 5 different colors; the pattern shows 5 is sufficient.
answered Feb 11 at 10:02
GlorfindelGlorfindel
14.4k45486
edited Feb 11 at 15:00
answered Feb 11 at 10:02
GlorfindelGlorfindel
14.4k45486
answered Feb 11 at 10:02
GlorfindelGlorfindel
14.4k45486
answered Feb 11 at 10:02
GlorfindelGlorfindel
14.4k45486
14.4k45486
$begingroup$
I think it's wrong, the angular triomino "2 5 3" (with the 5 at the angle) is repeating. The 2 on first line, 2nd column, and the 2 on 4th line 3rd column, they make the same triomino with the same colours.
$endgroup$
– Rémi Henry
Feb 11 at 10:09
$begingroup$
They are all repeating (since it's a pattern); the point is that each triomino has three different colours. Triominos are not compared with each other (at least that's how I interpreted the puzzle).
$endgroup$
– Glorfindel
Feb 11 at 10:11
1
$begingroup$
@Oray what's wrong with just extending the pattern?
$endgroup$
– Glorfindel
Feb 11 at 10:39
1
$begingroup$
If at all possible, it's good to use something else than different shades of red and green for conveying significant information.
$endgroup$
– Bass
Feb 11 at 11:23
1
$begingroup$
Yeah, good idea.
$endgroup$
– Glorfindel
Feb 11 at 11:31
|
show 7 more comments
$begingroup$
I think it's wrong, the angular triomino "2 5 3" (with the 5 at the angle) is repeating. The 2 on first line, 2nd column, and the 2 on 4th line 3rd column, they make the same triomino with the same colours.
$endgroup$
– Rémi Henry
Feb 11 at 10:09
$begingroup$
They are all repeating (since it's a pattern); the point is that each triomino has three different colours. Triominos are not compared with each other (at least that's how I interpreted the puzzle).
$endgroup$
– Glorfindel
Feb 11 at 10:11
1
$begingroup$
@Oray what's wrong with just extending the pattern?
$endgroup$
– Glorfindel
Feb 11 at 10:39
1
$begingroup$
If at all possible, it's good to use something else than different shades of red and green for conveying significant information.
$endgroup$
– Bass
Feb 11 at 11:23
1
$begingroup$
Yeah, good idea.
$endgroup$
– Glorfindel
Feb 11 at 11:31
$begingroup$
I think it's wrong, the angular triomino "2 5 3" (with the 5 at the angle) is repeating. The 2 on first line, 2nd column, and the 2 on 4th line 3rd column, they make the same triomino with the same colours.
$endgroup$
– Rémi Henry
Feb 11 at 10:09
$begingroup$
I think it's wrong, the angular triomino "2 5 3" (with the 5 at the angle) is repeating. The 2 on first line, 2nd column, and the 2 on 4th line 3rd column, they make the same triomino with the same colours.
$endgroup$
– Rémi Henry
Feb 11 at 10:09
$begingroup$
They are all repeating (since it's a pattern); the point is that each triomino has three different colours. Triominos are not compared with each other (at least that's how I interpreted the puzzle).
$endgroup$
– Glorfindel
Feb 11 at 10:11
$begingroup$
They are all repeating (since it's a pattern); the point is that each triomino has three different colours. Triominos are not compared with each other (at least that's how I interpreted the puzzle).
$endgroup$
– Glorfindel
Feb 11 at 10:11
1
1
$begingroup$
@Oray what's wrong with just extending the pattern?
$endgroup$
– Glorfindel
Feb 11 at 10:39
$begingroup$
@Oray what's wrong with just extending the pattern?
$endgroup$
– Glorfindel
Feb 11 at 10:39
1
1
$begingroup$
If at all possible, it's good to use something else than different shades of red and green for conveying significant information.
$endgroup$
– Bass
Feb 11 at 11:23
$begingroup$
If at all possible, it's good to use something else than different shades of red and green for conveying significant information.
$endgroup$
– Bass
Feb 11 at 11:23
1
1
$begingroup$
Yeah, good idea.
$endgroup$
– Glorfindel
Feb 11 at 11:31
$begingroup$
Yeah, good idea.
$endgroup$
– Glorfindel
Feb 11 at 11:31
|
show 7 more comments
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$begingroup$
Re: the edits, I was sure that the right name was triomino (based on tri(o)-), but I had put the word "popular" right before "science" because I wanted to stress that the magazine deals with popular science, so I don't think it necessarily breaks the adjective order rule.
$endgroup$
– Nautilus
Feb 11 at 14:41