Rectangles in a chess board
9
$begingroup$
How many rectangles can be made from the individual spaces of a chess board?
mathematics combinatorics
edited Feb 13 at 18:05
Glorfindel
13.8k35084
asked Feb 13 at 17:18
Dicul SmerdDicul Smerd
1
$endgroup$
|
show 3 more comments
9
$begingroup$
How many rectangles can be made from the individual spaces of a chess board?
mathematics combinatorics
edited Feb 13 at 18:05
Glorfindel
13.8k35084
asked Feb 13 at 17:18
Dicul SmerdDicul Smerd
1
$endgroup$
$begingroup$
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♦
Feb 13 at 17:50
$begingroup$
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♦
Feb 13 at 17:52
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
Feb 13 at 17:55
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
Feb 13 at 17:55
1
$begingroup$
No problem, just trying to help out
$endgroup$
– Greg
Feb 13 at 18:15
|
show 3 more comments
9
9
9
4
$begingroup$
How many rectangles can be made from the individual spaces of a chess board?
mathematics combinatorics
edited Feb 13 at 18:05
Glorfindel
13.8k35084
asked Feb 13 at 17:18
Dicul SmerdDicul Smerd
1
$endgroup$
How many rectangles can be made from the individual spaces of a chess board?
mathematics combinatorics
mathematics combinatorics
edited Feb 13 at 18:05
Glorfindel
13.8k35084
asked Feb 13 at 17:18
Dicul SmerdDicul Smerd
1
edited Feb 13 at 18:05
Glorfindel
13.8k35084
asked Feb 13 at 17:18
Dicul SmerdDicul Smerd
1
edited Feb 13 at 18:05
Glorfindel
13.8k35084
edited Feb 13 at 18:05
Glorfindel
13.8k35084
edited Feb 13 at 18:05
Glorfindel
13.8k35084
13.8k35084
asked Feb 13 at 17:18
Dicul SmerdDicul Smerd
1
asked Feb 13 at 17:18
Dicul SmerdDicul Smerd
1
asked Feb 13 at 17:18
Dicul SmerdDicul Smerd
1
1
$begingroup$
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♦
Feb 13 at 17:50
$begingroup$
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♦
Feb 13 at 17:52
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
Feb 13 at 17:55
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
Feb 13 at 17:55
1
$begingroup$
No problem, just trying to help out
$endgroup$
– Greg
Feb 13 at 18:15
|
show 3 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♦
Feb 13 at 17:50
$begingroup$
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♦
Feb 13 at 17:52
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
Feb 13 at 17:55
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
Feb 13 at 17:55
1
$begingroup$
No problem, just trying to help out
$endgroup$
– Greg
Feb 13 at 18:15
$begingroup$
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♦
Feb 13 at 17:50
$begingroup$
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♦
Feb 13 at 17:50
$begingroup$
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♦
Feb 13 at 17:52
$begingroup$
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♦
Feb 13 at 17:52
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
Feb 13 at 17:55
$begingroup$
@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
Feb 13 at 17:55
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
Feb 13 at 17:55
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
Feb 13 at 17:55
1
1
$begingroup$
No problem, just trying to help out
$endgroup$
– Greg
Feb 13 at 18:15
$begingroup$
No problem, just trying to help out
$endgroup$
– Greg
Feb 13 at 18:15
|
show 3 more comments
2 Answers
2
active
oldest
votes
15
$begingroup$
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 Feb 13 at 17:53
Gareth McCaughan♦Gareth McCaughan
63k3162246
$endgroup$
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
Feb 13 at 17:58
$begingroup$
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
Feb 13 at 21:51
$begingroup$
@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
Feb 14 at 0:12
$begingroup$
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♦
Feb 14 at 0:47
$begingroup$
Actually, it might not be that complex. A valid rectangle shape would be one where the slope of each side is an integer ratio, where one dimension is a multiple of the other (including a multiple of 1, which would be square), and where the sums of the two side lengths' rises and runs are <= board size. Then simply multiply by board size - the difference between those values (in each dimension) for transpositions.
$endgroup$
– Darrel Hoffman
Feb 14 at 18:01
|
show 1 more comment
11
$begingroup$
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 Feb 13 at 18:20
Dr XorileDr Xorile
12.5k22569
$endgroup$
3
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
Feb 13 at 18:36
$begingroup$
Aren't you.....
$endgroup$
– Dicul Smerd
Feb 14 at 8:33
$begingroup$
I am. Or at least that's what I studied. I don't get to do too much now. Why?
$endgroup$
– Dr Xorile
Feb 14 at 15:16
1
$begingroup$
I love physicists. Thinking you are one of them
$endgroup$
– Dicul Smerd
Feb 14 at 16:34
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "559"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fpuzzling.stackexchange.com%2fquestions%2f79653%2frectangles-in-a-chess-board%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
15
$begingroup$
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 Feb 13 at 17:53
Gareth McCaughan♦Gareth McCaughan
63k3162246
$endgroup$
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
Feb 13 at 17:58
$begingroup$
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
Feb 13 at 21:51
$begingroup$
@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
Feb 14 at 0:12
$begingroup$
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♦
Feb 14 at 0:47
$begingroup$
Actually, it might not be that complex. A valid rectangle shape would be one where the slope of each side is an integer ratio, where one dimension is a multiple of the other (including a multiple of 1, which would be square), and where the sums of the two side lengths' rises and runs are <= board size. Then simply multiply by board size - the difference between those values (in each dimension) for transpositions.
$endgroup$
– Darrel Hoffman
Feb 14 at 18:01
|
show 1 more comment
15
$begingroup$
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 Feb 13 at 17:53
Gareth McCaughan♦Gareth McCaughan
63k3162246
$endgroup$
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
Feb 13 at 17:58
$begingroup$
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
Feb 13 at 21:51
$begingroup$
@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
Feb 14 at 0:12
$begingroup$
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♦
Feb 14 at 0:47
$begingroup$
Actually, it might not be that complex. A valid rectangle shape would be one where the slope of each side is an integer ratio, where one dimension is a multiple of the other (including a multiple of 1, which would be square), and where the sums of the two side lengths' rises and runs are <= board size. Then simply multiply by board size - the difference between those values (in each dimension) for transpositions.
$endgroup$
– Darrel Hoffman
Feb 14 at 18:01
|
show 1 more comment
15
15
15
$begingroup$
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 Feb 13 at 17:53
Gareth McCaughan♦Gareth McCaughan
63k3162246
$endgroup$
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 Feb 13 at 17:53
Gareth McCaughan♦Gareth McCaughan
63k3162246
answered Feb 13 at 17:53
Gareth McCaughan♦Gareth McCaughan
63k3162246
answered Feb 13 at 17:53
Gareth McCaughan♦Gareth McCaughan
63k3162246
answered Feb 13 at 17:53
Gareth McCaughan♦Gareth McCaughan
63k3162246
63k3162246
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
Feb 13 at 17:58
$begingroup$
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
Feb 13 at 21:51
$begingroup$
@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
Feb 14 at 0:12
$begingroup$
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♦
Feb 14 at 0:47
$begingroup$
Actually, it might not be that complex. A valid rectangle shape would be one where the slope of each side is an integer ratio, where one dimension is a multiple of the other (including a multiple of 1, which would be square), and where the sums of the two side lengths' rises and runs are <= board size. Then simply multiply by board size - the difference between those values (in each dimension) for transpositions.
$endgroup$
– Darrel Hoffman
Feb 14 at 18:01
|
show 1 more comment
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
Feb 13 at 17:58
$begingroup$
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
Feb 13 at 21:51
$begingroup$
@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
Feb 14 at 0:12
$begingroup$
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♦
Feb 14 at 0:47
$begingroup$
Actually, it might not be that complex. A valid rectangle shape would be one where the slope of each side is an integer ratio, where one dimension is a multiple of the other (including a multiple of 1, which would be square), and where the sums of the two side lengths' rises and runs are <= board size. Then simply multiply by board size - the difference between those values (in each dimension) for transpositions.
$endgroup$
– Darrel Hoffman
Feb 14 at 18:01
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
Feb 13 at 17:58
$begingroup$
It is amazing that this hasn't been done already on PSE!
$endgroup$
– Dr Xorile
Feb 13 at 17:58
$begingroup$
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
Feb 13 at 21:51
$begingroup$
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
Feb 13 at 21:51
$begingroup$
@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
Feb 14 at 0:12
$begingroup$
@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
Feb 14 at 0:12
$begingroup$
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♦
Feb 14 at 0:47
$begingroup$
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♦
Feb 14 at 0:47
$begingroup$
Actually, it might not be that complex. A valid rectangle shape would be one where the slope of each side is an integer ratio, where one dimension is a multiple of the other (including a multiple of 1, which would be square), and where the sums of the two side lengths' rises and runs are <= board size. Then simply multiply by board size - the difference between those values (in each dimension) for transpositions.
$endgroup$
– Darrel Hoffman
Feb 14 at 18:01
$begingroup$
Actually, it might not be that complex. A valid rectangle shape would be one where the slope of each side is an integer ratio, where one dimension is a multiple of the other (including a multiple of 1, which would be square), and where the sums of the two side lengths' rises and runs are <= board size. Then simply multiply by board size - the difference between those values (in each dimension) for transpositions.
$endgroup$
– Darrel Hoffman
Feb 14 at 18:01
|
show 1 more comment
11
$begingroup$
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 Feb 13 at 18:20
Dr XorileDr Xorile
12.5k22569
$endgroup$
3
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
Feb 13 at 18:36
$begingroup$
Aren't you.....
$endgroup$
– Dicul Smerd
Feb 14 at 8:33
$begingroup$
I am. Or at least that's what I studied. I don't get to do too much now. Why?
$endgroup$
– Dr Xorile
Feb 14 at 15:16
1
$begingroup$
I love physicists. Thinking you are one of them
$endgroup$
– Dicul Smerd
Feb 14 at 16:34
add a comment |
11
$begingroup$
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 Feb 13 at 18:20
Dr XorileDr Xorile
12.5k22569
$endgroup$
3
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
Feb 13 at 18:36
$begingroup$
Aren't you.....
$endgroup$
– Dicul Smerd
Feb 14 at 8:33
$begingroup$
I am. Or at least that's what I studied. I don't get to do too much now. Why?
$endgroup$
– Dr Xorile
Feb 14 at 15:16
1
$begingroup$
I love physicists. Thinking you are one of them
$endgroup$
– Dicul Smerd
Feb 14 at 16:34
add a comment |
11
11
11
$begingroup$
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 Feb 13 at 18:20
Dr XorileDr Xorile
12.5k22569
$endgroup$
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 Feb 13 at 18:20
Dr XorileDr Xorile
12.5k22569
answered Feb 13 at 18:20
Dr XorileDr Xorile
12.5k22569
answered Feb 13 at 18:20
Dr XorileDr Xorile
12.5k22569
answered Feb 13 at 18:20
Dr XorileDr Xorile
12.5k22569
12.5k22569
3
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
Feb 13 at 18:36
$begingroup$
Aren't you.....
$endgroup$
– Dicul Smerd
Feb 14 at 8:33
$begingroup$
I am. Or at least that's what I studied. I don't get to do too much now. Why?
$endgroup$
– Dr Xorile
Feb 14 at 15:16
1
$begingroup$
I love physicists. Thinking you are one of them
$endgroup$
– Dicul Smerd
Feb 14 at 16:34
add a comment |
3
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
Feb 13 at 18:36
$begingroup$
Aren't you.....
$endgroup$
– Dicul Smerd
Feb 14 at 8:33
$begingroup$
I am. Or at least that's what I studied. I don't get to do too much now. Why?
$endgroup$
– Dr Xorile
Feb 14 at 15:16
1
$begingroup$
I love physicists. Thinking you are one of them
$endgroup$
– Dicul Smerd
Feb 14 at 16:34
3
3
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
Feb 13 at 18:36
$begingroup$
You are physicist right?
$endgroup$
– Dicul Smerd
Feb 13 at 18:36
$begingroup$
Aren't you.....
$endgroup$
– Dicul Smerd
Feb 14 at 8:33
$begingroup$
Aren't you.....
$endgroup$
– Dicul Smerd
Feb 14 at 8:33
$begingroup$
I am. Or at least that's what I studied. I don't get to do too much now. Why?
$endgroup$
– Dr Xorile
Feb 14 at 15:16
$begingroup$
I am. Or at least that's what I studied. I don't get to do too much now. Why?
$endgroup$
– Dr Xorile
Feb 14 at 15:16
1
1
$begingroup$
I love physicists. Thinking you are one of them
$endgroup$
– Dicul Smerd
Feb 14 at 16:34
$begingroup$
I love physicists. Thinking you are one of them
$endgroup$
– Dicul Smerd
Feb 14 at 16:34
add a comment |
draft saved
draft discarded
Thanks for contributing an answer to Puzzling Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fpuzzling.stackexchange.com%2fquestions%2f79653%2frectangles-in-a-chess-board%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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♦
Feb 13 at 17:50
$begingroup$
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♦
Feb 13 at 17:52
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
Feb 13 at 17:55
$begingroup$
Ah, OK. Hadn't noticed that the question had changed.
$endgroup$
– Gareth McCaughan♦
Feb 13 at 17:55
1
$begingroup$
No problem, just trying to help out
$endgroup$
– Greg
Feb 13 at 18:15