Rectangles in a chess board
7
$begingroup$
How many rectangles can be made from the individual spaces of a chess board?
mathematics combinatorics
edited 13 hours ago
Glorfindel
13.8k35084
asked 14 hours ago
Dicul SmerdDicul Smerd
4119
$endgroup$
|
show 2 more comments
7
$begingroup$
How many rectangles can be made from the individual spaces of a chess board?
mathematics combinatorics
edited 13 hours ago
Glorfindel
13.8k35084
asked 14 hours ago
Dicul SmerdDicul Smerd
4119
$endgroup$
$begingroup$
I don't think it's a duplicate of that, because it's asking about rectangles and not just squares. I'll be quite surprised if the rectangle question isn't already on PSE, though.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
OK, I think I'm surprised: I don't see any sign that this question has been asked here before. It's a bit of a maths-homework question, but I guess just about enough not so that I'm not about to close it.
$endgroup$
– Gareth McCaughan♦
13 hours ago
1
$begingroup$
@GarethMcCaughan, the question was originally about squares, but changed after I flagged it (see the edit). But yes, rectangles are different so I'll retract my flag
$endgroup$
– Greg
13 hours ago
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
Dicul, in the future, if you change your puzzle after it's been flagged, let the flagger know. I would have retracted the flag earlier
$endgroup$
– Greg
13 hours ago
|
show 2 more comments
7
7
7
4
$begingroup$
How many rectangles can be made from the individual spaces of a chess board?
mathematics combinatorics
edited 13 hours ago
Glorfindel
13.8k35084
asked 14 hours ago
Dicul SmerdDicul Smerd
4119
$endgroup$
How many rectangles can be made from the individual spaces of a chess board?
mathematics combinatorics
mathematics combinatorics
edited 13 hours ago
Glorfindel
13.8k35084
asked 14 hours ago
Dicul SmerdDicul Smerd
4119
edited 13 hours ago
Glorfindel
13.8k35084
asked 14 hours ago
Dicul SmerdDicul Smerd
4119
edited 13 hours ago
Glorfindel
13.8k35084
edited 13 hours ago
Glorfindel
13.8k35084
edited 13 hours ago
Glorfindel
13.8k35084
13.8k35084
asked 14 hours ago
Dicul SmerdDicul Smerd
4119
asked 14 hours ago
Dicul SmerdDicul Smerd
4119
asked 14 hours ago
Dicul SmerdDicul Smerd
4119
4119
$begingroup$
I don't think it's a duplicate of that, because it's asking about rectangles and not just squares. I'll be quite surprised if the rectangle question isn't already on PSE, though.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
OK, I think I'm surprised: I don't see any sign that this question has been asked here before. It's a bit of a maths-homework question, but I guess just about enough not so that I'm not about to close it.
$endgroup$
– Gareth McCaughan♦
13 hours ago
1
$begingroup$
@GarethMcCaughan, the question was originally about squares, but changed after I flagged it (see the edit). But yes, rectangles are different so I'll retract my flag
$endgroup$
– Greg
13 hours ago
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
Dicul, in the future, if you change your puzzle after it's been flagged, let the flagger know. I would have retracted the flag earlier
$endgroup$
– Greg
13 hours ago
|
show 2 more comments
$begingroup$
I don't think it's a duplicate of that, because it's asking about rectangles and not just squares. I'll be quite surprised if the rectangle question isn't already on PSE, though.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
OK, I think I'm surprised: I don't see any sign that this question has been asked here before. It's a bit of a maths-homework question, but I guess just about enough not so that I'm not about to close it.
$endgroup$
– Gareth McCaughan♦
13 hours ago
1
$begingroup$
@GarethMcCaughan, the question was originally about squares, but changed after I flagged it (see the edit). But yes, rectangles are different so I'll retract my flag
$endgroup$
– Greg
13 hours ago
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
Dicul, in the future, if you change your puzzle after it's been flagged, let the flagger know. I would have retracted the flag earlier
$endgroup$
– Greg
13 hours ago
$begingroup$
I don't think it's a duplicate of that, because it's asking about rectangles and not just squares. I'll be quite surprised if the rectangle question isn't already on PSE, though.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
I don't think it's a duplicate of that, because it's asking about rectangles and not just squares. I'll be quite surprised if the rectangle question isn't already on PSE, though.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
OK, I think I'm surprised: I don't see any sign that this question has been asked here before. It's a bit of a maths-homework question, but I guess just about enough not so that I'm not about to close it.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
OK, I think I'm surprised: I don't see any sign that this question has been asked here before. It's a bit of a maths-homework question, but I guess just about enough not so that I'm not about to close it.
$endgroup$
– Gareth McCaughan♦
13 hours ago
1
1
$begingroup$
@GarethMcCaughan, the question was originally about squares, but changed after I flagged it (see the edit). But yes, rectangles are different so I'll retract my flag
$endgroup$
– Greg
13 hours ago
$begingroup$
@GarethMcCaughan, the question was originally about squares, but changed after I flagged it (see the edit). But yes, rectangles are different so I'll retract my flag
$endgroup$
– Greg
13 hours ago
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
Dicul, in the future, if you change your puzzle after it's been flagged, let the flagger know. I would have retracted the flag earlier
$endgroup$
– Greg
13 hours ago
$begingroup$
Dicul, in the future, if you change your puzzle after it's been flagged, let the flagger know. I would have retracted the flag earlier
$endgroup$
– Greg
13 hours ago
|
show 2 more comments
2 Answers
2
active
oldest
votes
13
$begingroup$
To specify a rectangle
it suffices to say where its left and right boundaries are, and where its top and bottom boundaries are. There are $binom92$ choices for each pair of boundaries and therefore $binom92^2$ rectangles. That is to say, 1296 rectangles.
answered 13 hours ago
Gareth McCaughan♦Gareth McCaughan
62.2k3160242
$endgroup$
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
13 hours ago
$begingroup$
Are we only counting rectangles whose edges are made from the lines on the board? Because if you were to count all the rectangles possible by connecting corners between the squares, there's a lot more of them...
$endgroup$
– Darrel Hoffman
9 hours ago
$begingroup$
@DarrelHoffman As per the question these rectangles must be formed from "spaces". Hence I would expect at minimum that no rectangle cuts through one of the spaces used to define it. (The question of how many rectangles can be formed from an arbitrary group of points is perhaps more interesting than this one, though.)
$endgroup$
– Brilliand
7 hours ago
$begingroup$
Yes, the question becomes quite different (and I think much harder -- I suspect there is then no "elegant" solution) if you allow non-aligned rectangles. @Brilliand The question didn't always include the wording about "spaces" and may not have done when Darrel Hoffman wrote his comment.
$endgroup$
– Gareth McCaughan♦
6 hours ago
add a comment |
9
$begingroup$
Of course @Gareth_McCaughan got this well-known puzzle immediately. But for people who aren't up on their combinatorics, here's the same calculation in a way that seems easier (at least to me).
- There are 9x9 = 81 corners.
- For each of these there are 8x8 = 64 corners that are not in the same row or column.
- Each pair of these makes a rectangle.
- But then each rectangle has been counted four times (you can have top-left and bottom-right or top-right and bottom-left and both of those can be done two ways)
- So the final answer is 81 x 64/4 = 1296
answered 13 hours ago
Dr XorileDr Xorile
12.3k22568
$endgroup$
2
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
13 hours ago
add a comment |
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2 Answers
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2 Answers
2
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13
$begingroup$
To specify a rectangle
it suffices to say where its left and right boundaries are, and where its top and bottom boundaries are. There are $binom92$ choices for each pair of boundaries and therefore $binom92^2$ rectangles. That is to say, 1296 rectangles.
answered 13 hours ago
Gareth McCaughan♦Gareth McCaughan
62.2k3160242
$endgroup$
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
13 hours ago
$begingroup$
Are we only counting rectangles whose edges are made from the lines on the board? Because if you were to count all the rectangles possible by connecting corners between the squares, there's a lot more of them...
$endgroup$
– Darrel Hoffman
9 hours ago
$begingroup$
@DarrelHoffman As per the question these rectangles must be formed from "spaces". Hence I would expect at minimum that no rectangle cuts through one of the spaces used to define it. (The question of how many rectangles can be formed from an arbitrary group of points is perhaps more interesting than this one, though.)
$endgroup$
– Brilliand
7 hours ago
$begingroup$
Yes, the question becomes quite different (and I think much harder -- I suspect there is then no "elegant" solution) if you allow non-aligned rectangles. @Brilliand The question didn't always include the wording about "spaces" and may not have done when Darrel Hoffman wrote his comment.
$endgroup$
– Gareth McCaughan♦
6 hours ago
add a comment |
13
$begingroup$
To specify a rectangle
it suffices to say where its left and right boundaries are, and where its top and bottom boundaries are. There are $binom92$ choices for each pair of boundaries and therefore $binom92^2$ rectangles. That is to say, 1296 rectangles.
answered 13 hours ago
Gareth McCaughan♦Gareth McCaughan
62.2k3160242
$endgroup$
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
13 hours ago
$begingroup$
Are we only counting rectangles whose edges are made from the lines on the board? Because if you were to count all the rectangles possible by connecting corners between the squares, there's a lot more of them...
$endgroup$
– Darrel Hoffman
9 hours ago
$begingroup$
@DarrelHoffman As per the question these rectangles must be formed from "spaces". Hence I would expect at minimum that no rectangle cuts through one of the spaces used to define it. (The question of how many rectangles can be formed from an arbitrary group of points is perhaps more interesting than this one, though.)
$endgroup$
– Brilliand
7 hours ago
$begingroup$
Yes, the question becomes quite different (and I think much harder -- I suspect there is then no "elegant" solution) if you allow non-aligned rectangles. @Brilliand The question didn't always include the wording about "spaces" and may not have done when Darrel Hoffman wrote his comment.
$endgroup$
– Gareth McCaughan♦
6 hours ago
add a comment |
13
13
13
$begingroup$
To specify a rectangle
it suffices to say where its left and right boundaries are, and where its top and bottom boundaries are. There are $binom92$ choices for each pair of boundaries and therefore $binom92^2$ rectangles. That is to say, 1296 rectangles.
answered 13 hours ago
Gareth McCaughan♦Gareth McCaughan
62.2k3160242
$endgroup$
To specify a rectangle
it suffices to say where its left and right boundaries are, and where its top and bottom boundaries are. There are $binom92$ choices for each pair of boundaries and therefore $binom92^2$ rectangles. That is to say, 1296 rectangles.
answered 13 hours ago
Gareth McCaughan♦Gareth McCaughan
62.2k3160242
answered 13 hours ago
Gareth McCaughan♦Gareth McCaughan
62.2k3160242
answered 13 hours ago
Gareth McCaughan♦Gareth McCaughan
62.2k3160242
answered 13 hours ago
Gareth McCaughan♦Gareth McCaughan
62.2k3160242
62.2k3160242
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
13 hours ago
$begingroup$
Are we only counting rectangles whose edges are made from the lines on the board? Because if you were to count all the rectangles possible by connecting corners between the squares, there's a lot more of them...
$endgroup$
– Darrel Hoffman
9 hours ago
$begingroup$
@DarrelHoffman As per the question these rectangles must be formed from "spaces". Hence I would expect at minimum that no rectangle cuts through one of the spaces used to define it. (The question of how many rectangles can be formed from an arbitrary group of points is perhaps more interesting than this one, though.)
$endgroup$
– Brilliand
7 hours ago
$begingroup$
Yes, the question becomes quite different (and I think much harder -- I suspect there is then no "elegant" solution) if you allow non-aligned rectangles. @Brilliand The question didn't always include the wording about "spaces" and may not have done when Darrel Hoffman wrote his comment.
$endgroup$
– Gareth McCaughan♦
6 hours ago
add a comment |
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
13 hours ago
$begingroup$
Are we only counting rectangles whose edges are made from the lines on the board? Because if you were to count all the rectangles possible by connecting corners between the squares, there's a lot more of them...
$endgroup$
– Darrel Hoffman
9 hours ago
$begingroup$
@DarrelHoffman As per the question these rectangles must be formed from "spaces". Hence I would expect at minimum that no rectangle cuts through one of the spaces used to define it. (The question of how many rectangles can be formed from an arbitrary group of points is perhaps more interesting than this one, though.)
$endgroup$
– Brilliand
7 hours ago
$begingroup$
Yes, the question becomes quite different (and I think much harder -- I suspect there is then no "elegant" solution) if you allow non-aligned rectangles. @Brilliand The question didn't always include the wording about "spaces" and may not have done when Darrel Hoffman wrote his comment.
$endgroup$
– Gareth McCaughan♦
6 hours ago
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
13 hours ago
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
13 hours ago
$begingroup$
Are we only counting rectangles whose edges are made from the lines on the board? Because if you were to count all the rectangles possible by connecting corners between the squares, there's a lot more of them...
$endgroup$
– Darrel Hoffman
9 hours ago
$begingroup$
Are we only counting rectangles whose edges are made from the lines on the board? Because if you were to count all the rectangles possible by connecting corners between the squares, there's a lot more of them...
$endgroup$
– Darrel Hoffman
9 hours ago
$begingroup$
@DarrelHoffman As per the question these rectangles must be formed from "spaces". Hence I would expect at minimum that no rectangle cuts through one of the spaces used to define it. (The question of how many rectangles can be formed from an arbitrary group of points is perhaps more interesting than this one, though.)
$endgroup$
– Brilliand
7 hours ago
$begingroup$
@DarrelHoffman As per the question these rectangles must be formed from "spaces". Hence I would expect at minimum that no rectangle cuts through one of the spaces used to define it. (The question of how many rectangles can be formed from an arbitrary group of points is perhaps more interesting than this one, though.)
$endgroup$
– Brilliand
7 hours ago
$begingroup$
Yes, the question becomes quite different (and I think much harder -- I suspect there is then no "elegant" solution) if you allow non-aligned rectangles. @Brilliand The question didn't always include the wording about "spaces" and may not have done when Darrel Hoffman wrote his comment.
$endgroup$
– Gareth McCaughan♦
6 hours ago
$begingroup$
Yes, the question becomes quite different (and I think much harder -- I suspect there is then no "elegant" solution) if you allow non-aligned rectangles. @Brilliand The question didn't always include the wording about "spaces" and may not have done when Darrel Hoffman wrote his comment.
$endgroup$
– Gareth McCaughan♦
6 hours ago
add a comment |
9
$begingroup$
Of course @Gareth_McCaughan got this well-known puzzle immediately. But for people who aren't up on their combinatorics, here's the same calculation in a way that seems easier (at least to me).
- There are 9x9 = 81 corners.
- For each of these there are 8x8 = 64 corners that are not in the same row or column.
- Each pair of these makes a rectangle.
- But then each rectangle has been counted four times (you can have top-left and bottom-right or top-right and bottom-left and both of those can be done two ways)
- So the final answer is 81 x 64/4 = 1296
answered 13 hours ago
Dr XorileDr Xorile
12.3k22568
$endgroup$
2
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
13 hours ago
add a comment |
9
$begingroup$
Of course @Gareth_McCaughan got this well-known puzzle immediately. But for people who aren't up on their combinatorics, here's the same calculation in a way that seems easier (at least to me).
- There are 9x9 = 81 corners.
- For each of these there are 8x8 = 64 corners that are not in the same row or column.
- Each pair of these makes a rectangle.
- But then each rectangle has been counted four times (you can have top-left and bottom-right or top-right and bottom-left and both of those can be done two ways)
- So the final answer is 81 x 64/4 = 1296
answered 13 hours ago
Dr XorileDr Xorile
12.3k22568
$endgroup$
2
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
13 hours ago
add a comment |
9
9
9
$begingroup$
Of course @Gareth_McCaughan got this well-known puzzle immediately. But for people who aren't up on their combinatorics, here's the same calculation in a way that seems easier (at least to me).
- There are 9x9 = 81 corners.
- For each of these there are 8x8 = 64 corners that are not in the same row or column.
- Each pair of these makes a rectangle.
- But then each rectangle has been counted four times (you can have top-left and bottom-right or top-right and bottom-left and both of those can be done two ways)
- So the final answer is 81 x 64/4 = 1296
answered 13 hours ago
Dr XorileDr Xorile
12.3k22568
$endgroup$
Of course @Gareth_McCaughan got this well-known puzzle immediately. But for people who aren't up on their combinatorics, here's the same calculation in a way that seems easier (at least to me).
- There are 9x9 = 81 corners.
- For each of these there are 8x8 = 64 corners that are not in the same row or column.
- Each pair of these makes a rectangle.
- But then each rectangle has been counted four times (you can have top-left and bottom-right or top-right and bottom-left and both of those can be done two ways)
- So the final answer is 81 x 64/4 = 1296
answered 13 hours ago
Dr XorileDr Xorile
12.3k22568
answered 13 hours ago
Dr XorileDr Xorile
12.3k22568
answered 13 hours ago
Dr XorileDr Xorile
12.3k22568
answered 13 hours ago
Dr XorileDr Xorile
12.3k22568
12.3k22568
2
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
13 hours ago
add a comment |
2
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
13 hours ago
2
2
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
13 hours ago
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
13 hours ago
add a comment |
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$begingroup$
I don't think it's a duplicate of that, because it's asking about rectangles and not just squares. I'll be quite surprised if the rectangle question isn't already on PSE, though.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
OK, I think I'm surprised: I don't see any sign that this question has been asked here before. It's a bit of a maths-homework question, but I guess just about enough not so that I'm not about to close it.
$endgroup$
– Gareth McCaughan♦
13 hours ago
1
$begingroup$
@GarethMcCaughan, the question was originally about squares, but changed after I flagged it (see the edit). But yes, rectangles are different so I'll retract my flag
$endgroup$
– Greg
13 hours ago
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
13 hours ago
$begingroup$
Dicul, in the future, if you change your puzzle after it's been flagged, let the flagger know. I would have retracted the flag earlier
$endgroup$
– Greg
13 hours ago