question on gluing many 1 x 1 squares to form a rectangle
0
$begingroup$
"You have many 1 x 1 unit squares. You may colour the edges with one of four colours and glue them together along edges of the same colour. Your aim is to get an m x n rectangle. For what value of m and n is it possible?"
I have the above question with me. Trying out a few squares is it not possible for any such m and n. I guess so!! Like I can take any two squares and glue them along the same colours and attach one more square having the required edge colour and glue it?
coloring
asked Dec 11 '18 at 10:01
saisanjeevsaisanjeev
967212
$endgroup$
|
show 2 more comments
0
$begingroup$
"You have many 1 x 1 unit squares. You may colour the edges with one of four colours and glue them together along edges of the same colour. Your aim is to get an m x n rectangle. For what value of m and n is it possible?"
I have the above question with me. Trying out a few squares is it not possible for any such m and n. I guess so!! Like I can take any two squares and glue them along the same colours and attach one more square having the required edge colour and glue it?
coloring
asked Dec 11 '18 at 10:01
saisanjeevsaisanjeev
967212
$endgroup$
1
$begingroup$
Something seems to be missing from the question. Can you not colour all the edges of the squares the same colour, and then glue the squares together in a rectangle as large as you want?
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 10:12
$begingroup$
It is not clear what the constraints are. Can you rotate the squares? can you flip them? Are you allowed to color each square individually or you are required to color all squares in same manner? Let's say you are required to color all squares in same manner and you are not allowed to rotate/flip the square, then you can't even glue two squares into a $1 times 2$ rectangle.
$endgroup$
– achille hui
Dec 11 '18 at 10:15
$begingroup$
yupp no such constraints given. The problem is from the book "Problem Solving Strategies" by Arthur Engel, page 27 question number 21
$endgroup$
– saisanjeev
Dec 11 '18 at 14:04
$begingroup$
A variant of the same question is also there in the above mentioned book, question number 22, wherein the same arrangement is extended to 3 dimension, squares replaced by cubes
$endgroup$
– saisanjeev
Dec 11 '18 at 14:06
1
$begingroup$
The question is indeed as quoted from the book, but it is missing information. From reading the answer, it seems that the squares need to have four differently coloured edges, and furthermore, the finished rectangle must also have four differently coloured sides (and each rectangle side is just one colour). None of this is stated in the question. The followup cube question at least states that each cube has six colours and that the final faces should all have a different colour.
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 14:53
|
show 2 more comments
0
0
0
$begingroup$
"You have many 1 x 1 unit squares. You may colour the edges with one of four colours and glue them together along edges of the same colour. Your aim is to get an m x n rectangle. For what value of m and n is it possible?"
I have the above question with me. Trying out a few squares is it not possible for any such m and n. I guess so!! Like I can take any two squares and glue them along the same colours and attach one more square having the required edge colour and glue it?
coloring
asked Dec 11 '18 at 10:01
saisanjeevsaisanjeev
967212
$endgroup$
"You have many 1 x 1 unit squares. You may colour the edges with one of four colours and glue them together along edges of the same colour. Your aim is to get an m x n rectangle. For what value of m and n is it possible?"
I have the above question with me. Trying out a few squares is it not possible for any such m and n. I guess so!! Like I can take any two squares and glue them along the same colours and attach one more square having the required edge colour and glue it?
coloring
coloring
asked Dec 11 '18 at 10:01
saisanjeevsaisanjeev
967212
asked Dec 11 '18 at 10:01
saisanjeevsaisanjeev
967212
asked Dec 11 '18 at 10:01
saisanjeevsaisanjeev
967212
asked Dec 11 '18 at 10:01
saisanjeevsaisanjeev
967212
asked Dec 11 '18 at 10:01
saisanjeevsaisanjeev
967212
967212
1
$begingroup$
Something seems to be missing from the question. Can you not colour all the edges of the squares the same colour, and then glue the squares together in a rectangle as large as you want?
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 10:12
$begingroup$
It is not clear what the constraints are. Can you rotate the squares? can you flip them? Are you allowed to color each square individually or you are required to color all squares in same manner? Let's say you are required to color all squares in same manner and you are not allowed to rotate/flip the square, then you can't even glue two squares into a $1 times 2$ rectangle.
$endgroup$
– achille hui
Dec 11 '18 at 10:15
$begingroup$
yupp no such constraints given. The problem is from the book "Problem Solving Strategies" by Arthur Engel, page 27 question number 21
$endgroup$
– saisanjeev
Dec 11 '18 at 14:04
$begingroup$
A variant of the same question is also there in the above mentioned book, question number 22, wherein the same arrangement is extended to 3 dimension, squares replaced by cubes
$endgroup$
– saisanjeev
Dec 11 '18 at 14:06
1
$begingroup$
The question is indeed as quoted from the book, but it is missing information. From reading the answer, it seems that the squares need to have four differently coloured edges, and furthermore, the finished rectangle must also have four differently coloured sides (and each rectangle side is just one colour). None of this is stated in the question. The followup cube question at least states that each cube has six colours and that the final faces should all have a different colour.
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 14:53
|
show 2 more comments
1
$begingroup$
Something seems to be missing from the question. Can you not colour all the edges of the squares the same colour, and then glue the squares together in a rectangle as large as you want?
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 10:12
$begingroup$
It is not clear what the constraints are. Can you rotate the squares? can you flip them? Are you allowed to color each square individually or you are required to color all squares in same manner? Let's say you are required to color all squares in same manner and you are not allowed to rotate/flip the square, then you can't even glue two squares into a $1 times 2$ rectangle.
$endgroup$
– achille hui
Dec 11 '18 at 10:15
$begingroup$
yupp no such constraints given. The problem is from the book "Problem Solving Strategies" by Arthur Engel, page 27 question number 21
$endgroup$
– saisanjeev
Dec 11 '18 at 14:04
$begingroup$
A variant of the same question is also there in the above mentioned book, question number 22, wherein the same arrangement is extended to 3 dimension, squares replaced by cubes
$endgroup$
– saisanjeev
Dec 11 '18 at 14:06
1
$begingroup$
The question is indeed as quoted from the book, but it is missing information. From reading the answer, it seems that the squares need to have four differently coloured edges, and furthermore, the finished rectangle must also have four differently coloured sides (and each rectangle side is just one colour). None of this is stated in the question. The followup cube question at least states that each cube has six colours and that the final faces should all have a different colour.
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 14:53
1
1
$begingroup$
Something seems to be missing from the question. Can you not colour all the edges of the squares the same colour, and then glue the squares together in a rectangle as large as you want?
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 10:12
$begingroup$
Something seems to be missing from the question. Can you not colour all the edges of the squares the same colour, and then glue the squares together in a rectangle as large as you want?
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 10:12
$begingroup$
It is not clear what the constraints are. Can you rotate the squares? can you flip them? Are you allowed to color each square individually or you are required to color all squares in same manner? Let's say you are required to color all squares in same manner and you are not allowed to rotate/flip the square, then you can't even glue two squares into a $1 times 2$ rectangle.
$endgroup$
– achille hui
Dec 11 '18 at 10:15
$begingroup$
It is not clear what the constraints are. Can you rotate the squares? can you flip them? Are you allowed to color each square individually or you are required to color all squares in same manner? Let's say you are required to color all squares in same manner and you are not allowed to rotate/flip the square, then you can't even glue two squares into a $1 times 2$ rectangle.
$endgroup$
– achille hui
Dec 11 '18 at 10:15
$begingroup$
yupp no such constraints given. The problem is from the book "Problem Solving Strategies" by Arthur Engel, page 27 question number 21
$endgroup$
– saisanjeev
Dec 11 '18 at 14:04
$begingroup$
yupp no such constraints given. The problem is from the book "Problem Solving Strategies" by Arthur Engel, page 27 question number 21
$endgroup$
– saisanjeev
Dec 11 '18 at 14:04
$begingroup$
A variant of the same question is also there in the above mentioned book, question number 22, wherein the same arrangement is extended to 3 dimension, squares replaced by cubes
$endgroup$
– saisanjeev
Dec 11 '18 at 14:06
$begingroup$
A variant of the same question is also there in the above mentioned book, question number 22, wherein the same arrangement is extended to 3 dimension, squares replaced by cubes
$endgroup$
– saisanjeev
Dec 11 '18 at 14:06
1
1
$begingroup$
The question is indeed as quoted from the book, but it is missing information. From reading the answer, it seems that the squares need to have four differently coloured edges, and furthermore, the finished rectangle must also have four differently coloured sides (and each rectangle side is just one colour). None of this is stated in the question. The followup cube question at least states that each cube has six colours and that the final faces should all have a different colour.
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 14:53
$begingroup$
The question is indeed as quoted from the book, but it is missing information. From reading the answer, it seems that the squares need to have four differently coloured edges, and furthermore, the finished rectangle must also have four differently coloured sides (and each rectangle side is just one colour). None of this is stated in the question. The followup cube question at least states that each cube has six colours and that the final faces should all have a different colour.
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 14:53
|
show 2 more comments
0
active
oldest
votes
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: "69"
};
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: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
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%2fmath.stackexchange.com%2fquestions%2f3035132%2fquestion-on-gluing-many-1-x-1-squares-to-form-a-rectangle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
draft saved
draft discarded
Thanks for contributing an answer to Mathematics 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%2fmath.stackexchange.com%2fquestions%2f3035132%2fquestion-on-gluing-many-1-x-1-squares-to-form-a-rectangle%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
1
$begingroup$
Something seems to be missing from the question. Can you not colour all the edges of the squares the same colour, and then glue the squares together in a rectangle as large as you want?
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 10:12
$begingroup$
It is not clear what the constraints are. Can you rotate the squares? can you flip them? Are you allowed to color each square individually or you are required to color all squares in same manner? Let's say you are required to color all squares in same manner and you are not allowed to rotate/flip the square, then you can't even glue two squares into a $1 times 2$ rectangle.
$endgroup$
– achille hui
Dec 11 '18 at 10:15
$begingroup$
yupp no such constraints given. The problem is from the book "Problem Solving Strategies" by Arthur Engel, page 27 question number 21
$endgroup$
– saisanjeev
Dec 11 '18 at 14:04
$begingroup$
A variant of the same question is also there in the above mentioned book, question number 22, wherein the same arrangement is extended to 3 dimension, squares replaced by cubes
$endgroup$
– saisanjeev
Dec 11 '18 at 14:06
1
$begingroup$
The question is indeed as quoted from the book, but it is missing information. From reading the answer, it seems that the squares need to have four differently coloured edges, and furthermore, the finished rectangle must also have four differently coloured sides (and each rectangle side is just one colour). None of this is stated in the question. The followup cube question at least states that each cube has six colours and that the final faces should all have a different colour.
$endgroup$
– Jaap Scherphuis
Dec 11 '18 at 14:53