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[[File:De Bruijn theorem coloring.svg|thumb|A coloring of the unit cubes in a <math>\scriptstyle 6\times 6\times 6</math> box that may be used to prove the impossibility of packing it with <math>\scriptstyle 1\times 2\times 4</math> bricks]]
In a 1969 paper, Dutch mathematician [[Nicolaas Govert de Bruijn]] proved several results about packing [[congruence (geometry)|congruent]] rectangular bricks (of any dimension) into larger rectangular boxes, in such a way that no space is left over. One of these results is now known as '''de Bruijn's theorem'''. According to this theorem, a "harmonic brick" (one in which each side length is a multiple of the next smaller side length) can only be packed into a box whose dimensions are multiples of the brick's dimensions.<ref name="db69">{{citation
| last = de Bruijn | first = N. G. | author-link = Nicolaas Govert de Bruijn
| doi = 10.2307/2316785
| journal = The American Mathematical Monthly
| mr = 0234841
| pages = 37–40
| title = Filling boxes with bricks
| volume = 76
| year = 1969}}.</ref>


==Example==
De Bruijn was led to prove this result after his then-seven-year-old son, F. W. de Bruijn, was unable to pack bricks of dimension <math>\scriptstyle 1\times 2\times 4</math> into a <math>\scriptstyle 6\times 6\times 6</math> cube.<ref>{{citation
| last = Honsberger | first = Ross
| isbn = 9780883853009
| location = Washington, DC
| page = 69
| publisher = Mathematical Association of America
| title = Mathematical Gems II
| year = 1976}}.</ref><ref>{{citation
| last = Nienhuys | first = J. W.
| editor1-last = Kloks | editor2-first = Ling-Ju |editor2-last = Hung | editor1-first = Ton
| date = September 11, 2011
| page = 156
| title = De Bruijn's combinatorics: classroom notes
| url = http://books.google.com/books?id=AsXUY604pewC&pg=PA156}}.</ref> The cube has a volume equal to that of <math>\scriptstyle 27</math> bricks, but only <math>\scriptstyle 26</math> bricks may be packed into it; one way to see this is to partition the cube into <math>\scriptstyle 27</math> smaller cubes colored alternately black and white, and to observe that this coloring has more unit cells of one color than of the other, whereas with this coloring any placement of the <math>\scriptstyle 1\times 2\times 4</math> brick must have equal numbers of cells of each color.<ref>{{citation
| last = Watkins | first = John J.
| isbn = 9781400840922
| page = 226
| publisher = Princeton University Press
| title = Across the Board: The Mathematics of Chessboard Problems
| url = http://books.google.com/books?id=LtoSZmGzVs4C&pg=PA226
| year = 2012}}.</ref> De Bruijn's theorem proves that a perfect packing with these dimensions is impossible, in a more general way that applies to many other dimensions of bricks and boxes.


==Boxes that are multiples of the brick==
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Let us operate collectively to hold the discussion civil. nnSubmit CommentnPost your commentnnWe inspire you to register/login and move forward.nHowever, if you select to remain non-loggedin, you are required to fill up the sort under and verify your e-mail deal with before we can publish your comment.nThat way, we can confirm that the e-mail deal with doesn't belong to someone else.nRegister/LoginnProceed With out RegistrationnnFull Name nnLocation nnEmail nnFix six + =2nnnnProceed nnPost your commentnnWe inspire you to sign-up/login and move forward.nHowever, if you decide on to continue to be non-loggedin, you are essential to fill up the kind underneath and verify your e-mail address ahead of we can publish your remark.nThat way, we can verify that the electronic mail tackle does not belong to a person else.nRegister/LoginnProceed With out RegistrationnnFull Title nnLocation nnEmail nnFix six + =twonnnnProceed nnnnnThanks! nnWe have despatched you a verification e mail.nTo confirm, just follow the hyperlink in the message nnResend verification mailEmail despatchednnnSort by:Newest|Oldest|Recommended|Most Mentioned|Agree|DisagreeConnect with usFEATURED Today IN US & Canada NewsIndian activist poised to be initial at any time elected socialist in USnIndian-American surgeon gets optimum US immigrant awardnHave world's best task, says US lawful star Preet BhararanPromoting Punjabi heritage in CanadanChhath Puja celebrated in USnmoreYOUR FRIENDS' ACTIVITY1of1FEATURED Today IN NRIIndian jailed in Moscow for $18 million fraudn3 of Indian-origin among Africa's fifty richest: ForbesnIndian origin sisters sentenced for theftnProtests in Uk in help of Indian demise row convict in UAEnIndo-Canadian lady alleges racism in Vancouver barn10 Indians arrested in Uk for immigration offencesn80-year-previous Sikh attacked by lady in British isles diesnRs 1 crore relief for kin of NRI killed in mishapn1 in 10 medical doctors practising in United kingdom is from IndianDo Indians turn out to be far more religious when overseas?nmoreTOP STORIESOnion rates up 278%, tomato 122% in a single yearnAt nine.30am these days, India will appear to a halt as Sachin resumes his innings at 38 in final TestnMeet the thirteen-yr-old lady from Lucknow who is pursuing MSc in microbiologynIndian poised to be very first elected socialist in USnVolvo hearth: Techie smashed pane with head, saved manynmoreMOST POPULARReadn|nCommentedn|nSharednFrom NRI3 of Indian-origin between Africa's 50 richest: ForbesnIndian activist poised to be very first at any time elected socialist in USnIndian-American surgeon gets greatest US immigrant awardnIndian origin sisters sentenced for theftnmoreAcross Moments of IndiaDoodle 4 Google winner's entry marks Children's DaynInd vs WI: Sachin Tendulkar takes India to 157/two at stumps on Day 1 of 2nd TestnRajiv Gandhi wished Bofors cash to run Congress: Ex-CBI chiefn'With Sachin Paaji, a portion of me will be long gone too'nmoreAllnTOInMy ProfilenSign into earn your BadgesKnow far more about Moments PointsTHE Instances OF INDIAnPowered by INDIATIMESnAbout usnnPromote with usnnTerms of Use and Grievance Redressal CoveragennPrivacy plannnFeedbacknnRSSnnNewsletternnTOI MobilennePapernnSitemapnnArchivesnOther Moments Team news sitesnTimes Crest | The Economic Timesn&#x907&#x915&#x928&#x949&#x92e&#x93f&#x915 &#x91f&#x93e&#x907&#x92e&#x94d&#x938 | &#xa88&#xa95&#xacb&#xaa8&#xacb&#xaae&#xabf&#xa95 &#xa9f&#xabe&#xa88&#xaae&#xacd&#xab8nPune Mirror | Bangalore MirrornAhmedabad Mirror | ItsMyAscentnEducation Occasions | Brand CapitalnMumbai Mirror | Times NownIndiatimes | &#x928&#x935&#x92d&#x93e&#x930&#x924 &#x91f&#x93e&#x907&#x92e&#x94d&#x938n&#x92e&#x939&#x93e&#x930&#x93e&#x937&#x94d&#x91f&#x94d&#x930 &#x91f&#x93e&#x907&#x92e&#x94d&#x938 | &#xcb5&#xcbf&#xc9c&#xcaf &#xc95&#xcb0&#xccd&#xca8&#xcbe&#xc9f&#xc95nGo GreenLiving and entertainmentnTimescity | iDiva | Bollywood | ZoomnHealthmeup | Luxpresso | TechnoholiknGuylife | On the internet SongsnnCuriosity Networknitimes | EmailHot on the WebnHotklix | World | PoliticsnBusiness | SportsnEntertainmentnScience & TechnNew CarsServicesnBook print ads | On-line purchasing | Cost-free SMS | Site design | CRM | TendersnMatrimonial | Ringtones | Astrology | Work | Tech Neighborhood | Residence | Acquire carnBikes in India | Offers | Cost-free Classifieds | Send out income to India | Utilised CarsnRestaurants in Delhi | Motion picture Display Timings in Mumbai | Remit to India | Buy MobilesnListen Tracks | Information | Real Estate DevelopersnnTrending TopicsnYahoo | Intercourse | You Tube | Gold Price tag | Twitter | Irctc | Renault Duster | View Videos | Aadhar Card | Facebook | UID CardCopyright � 2013 Bennett, Coleman & Co. Ltd. All legal rights reserved. For reprint legal rights: Moments Syndication ServiceFind this comment offensive?nChoose your reason beneath and click [http://tinyurl.com/pch83be uggs on sale] the Submit button. This will notify our moderators to consider actionnNamennExplanation for reporting:nFoul languageSlanderousInciting hatred in opposition to a particular communitySpam/out of contextOthersnnYour Explanation has been Submitted to the admin.nnWhile observing her off to Canada, minor did her neighbours think about that the girl-up coming-doorway in Chandigarh, Guddu, would turn into RJ Aman, and make an identification as a woman aggressively operating to encourage Punjabi heritage amid NRIs settled in Canad<br><br>
Suppose that a <math>\scriptstyle d</math>-dimensional rectangular box (mathematically a [[cuboid]]) has [[integer]] side lengths <math>\scriptstyle A_1 \times A_2 \times \dots \times A_d</math> and a brick has lengths <math>\scriptstyle a_1 \times a_2 \times \dots \times a_d</math>. If the sides of the brick can be multiplied by another set of integers <math>\scriptstyle b_i</math> so that <math>\scriptstyle a_1b_1,a_2b_2,\dots a_d b_d</math> are a [[permutation]] of <math>\scriptstyle A_1, A_2,\dots,A_d</math>, the box is called a ''multiple'' of the brick. The box can then be filled with such bricks in a trivial way with all the bricks oriented the same way.<ref name="db69"/>
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==A generalization==
Not every packing involves boxes that are multiples of bricks. For instance, as de Bruijn observes, a <math>\scriptstyle 5\times 6</math> rectangular box can be filled with copies of a <math>\scriptstyle 2\times 3</math> rectangular brick, although not with all the bricks oriented the same way. However, {{harvtxt|de Bruijn|1969}} proves that if the bricks can fill the box, then for each <math>a_i,</math> at least one of the <math>A_i</math> is a multiple. In the above example, the side of length <math>6</math> is a multiple of both <math>2</math> and <math>3</math>.<ref name="db69"/>
 
==Harmonic bricks==
The second of de Bruijn's results, the one called de Bruijn's theorem, concerns the case where each side of the brick is a multiple of the next smaller side. De Bruijn calls a brick with this property ''harmonic''. For instance, the most frequently used [[brick]]s in the USA have dimensions <math>\scriptstyle 2\frac{1}{4}\times 4\times 8</math> (in inches), which is not harmonic, but a type of brick sold as "Roman brick" has the harmonic dimensions <math>\scriptstyle 2\times 4\times 12</math>.<ref>{{citation
| last = Kreh | first = R. T.
| edition = 5th
| isbn = 9780766859364
| page = 18
| publisher = Cengage Learning
| title = Masonry Skills
| url = http://books.google.com/books?id=e3gyN-TPRd4C&pg=PA18
| year = 2003}}.</ref>
 
De Bruijn's theorem states that, if a harmonic brick is packed into a box, then the box must be a multiple of the brick. For instance, the three-dimensional harmonic brick with side lengths 1, 2, and 6 can only be packed into boxes in which one of the three sides is a multiple of six and one of the remaining two sides is even.<ref name="db69"/><ref>{{citation
| last1 = Stein | first1 = Sherman K.
| last2 = Szabó | first2 = Sándor
| isbn = 0-88385-028-1
| location = Washington, DC
| mr = 1311249
| page = 52
| publisher = Mathematical Association of America
| series = Carus Mathematical Monographs
| title = Algebra and Tiling: Homomorphisms in the Service of Geometry
| url = http://books.google.com/books?id=QOa-mnX5Y4QC&pg=PA52
| volume = 25
| year = 1994}}.</ref> Packings of a harmonic brick into a box may involve copies of the brick that are rotated with respect to each other. Nevertheless, the theorem states that the only boxes that can be packed in this way are boxes that could also be packed by translates of the brick.
 
{{harvtxt|Boisen|1995}} provided an alternative proof of the three-dimensional case of de Bruijn's theorem, based on the algebra of [[polynomial]]s.<ref>{{citation
| last = Boisen | first = Paul
| doi = 10.1016/0012-365X(94)00070-1
| issue = 1-3
| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| mr = 1360122
| pages = 285–287
| title = Polynomials and packings: a new proof of de Bruijn's theorem
| volume = 146
| year = 1995}}.</ref>
 
==Non-harmonic bricks==
The third of de Bruijn's results is that, if a brick is not harmonic, then there is a box that it can fill that is not a multiple of the brick. The packing of the <math>\scriptstyle 2\times 3</math> brick into the <math>\scriptstyle 5\times 6</math> box provides an example of this phenomenon.<ref name="db69"/>
 
[[File:7x10 box packed with 2x5 bricks.svg|thumb|An <math>\scriptstyle (a_1+a_2)\times (a_1a_2)</math> box, tiled with <math>\scriptstyle a_1\times a_2</math> bricks, for the case <math>\scriptstyle a_1=2</math> and <math>\scriptstyle a_2=5</math>]]
In the two-dimensional case, the third of de Bruijn's results is easy to visualize. A box with dimensions <math>\scriptstyle A_1=a_1</math> and <math>\scriptstyle A_2=a_1a_2</math> is easy to pack with <math>\scriptstyle a_1</math> copies of a brick with dimensions <math>\scriptstyle a_1,a_2</math>, placed side by side. For the same reason, a box with dimensions <math>\scriptstyle A_1=a_1a_2</math> and <math>\scriptstyle A_2=a_2</math> is also easy to pack with copies of the same brick. Rotating one of these two boxes so that their long sides are parallel and placing them side by side results in a packing of a larger box with <math>\scriptstyle A_1=a_1+a_2</math> and <math>\scriptstyle A_2=a_1a_2</math>. This larger box is a multiple of the brick if and only if the brick is harmonic.
 
==References==
{{reflist}}
 
==External links==
*{{mathworld|title=de Bruijn's Theorem|urlname=deBruijnsTheorem}}
 
[[Category:Theorems in discrete geometry]]

Revision as of 06:01, 20 February 2014


CHANDIGARH: Whilst observing her off to Canada, little did her neighbours envision that the female-subsequent-door in Chandigarh, Guddu, would become RJ Aman, and receive an identification as a female aggressively doing work to promote Punjabi heritage amongst NRIs settled in Canada. nnIn her late 30s, and the mother of two sons - Rishi (fourteen) and Chirag (12) - Aman and her partner Sunil Dhariwal, who are settled in Mississauga, have extended tales to notify about their wrestle in Canada in which they landed in 1993. nnInitially, she experienced migrated to Canada with her http://tinyurl.com/pch83be dad and mom Jagjit Singh Rishi and Gurdev Kaur Rishi, a center course pair from Sector eight, Chandigarh. nnA graduate from the Federal government Higher education for Women, Sector ugg boots sale 11, Chandigarh, Aman did her masters in organization, and is entrepreneur-turned-radio jockey. nnAman runs her weekly radio programme referred to as 'Sadda ugg boots usa Virsa - http://tinyurl.com/pch83be Our Heritage' on Apna Radio in Canada.nn"In the long run, we are Punjabis, and our children need to know what our society is and how wealthy our heritage is. Originally, we utilised to talk about our culture and heritage in our social gatherings. My family and pals inspired me to commence some thing on these strains. Considering that I had some expertise in the university times, I opted for a radio present and began as a co-host. For the past close to two years, I have my own show. Now, not just my youngsters, but the youngsters of all Punjabis - like these from Pakistan and some other locations - interact and sing along," she added. nn"Apart from Full Masti and Gup-shup, this programme is my humble endeavor to remain in contact with the sturdy roots that India and Pakistan have supplied, besides celebrating the wings that our new property - Canada has given us. Naal Virse de Jurr ke rahange, for every ethe di vi gal karange ("We shall talk about Canada, but not at the cost of our wealthy heritage"), said Aman. nn"Our exhibits encourage each other to be better and do better. We draw inspiration from our scriptures - be it Shabad, Bhajan, Bible verses, Quran di aayat, or motivational prices," she insisted. nnAman chats with experts from in the community to learn their secrets and techniques of accomplishment, which in change helps everybody increase their skills and better combine into the mainstream neighborhood. She provides ahead concerns that pull the local community back again and hinder its development - be it neglect of elders, woman foeticide, domestic violence, youth and medicines, bullying in educational institutions, or mental well being. nnPut up a commentALSO ON TOInAAP is absolutely nothing but abusing and...nnPolitics proceeds more than Ashok ...nnDelhi polls: Sheila Dikshit fi...nn12 injured in explosion inside of...nnModi hits back at tea-vendor r...nnBJP slams Naresh Agarwal for '...nnView: Extraordinary accident caugh...nnOregon blast referred to as 'domestic ...nnVery first visuals: Terror hideout ...nnMuzaffarnagar riots: Two BJP M...nnCondominium hearth leaves Arizona ...nnVirginia: Air Pressure officer no...nnDelhi polls: Severe Vardhan fil...nnBihar: Violence in Danapur ove...nnSignificantly less than 27,000 signed up on ...nFollow the Instances of India - NRI sectionSearch for News SearchPost yet another commentnShare your sights - submit your remark under nnWe inspire you to log in and remark. Non-logged-in feedback will go through a verification method for safety factors.nnLog in to publish this commentnnYour commentnnCharacters Remaining: 3000nnRefrain from submitting remarks that are obscene, libellous, slanderous or inflammatory, and do not indulge in personalized attacks, identify contacting or inciting hatred against any group. Help us delete responses that do not follow these recommendations by marking them offensive. Let us operate collectively to hold the discussion civil. nnSubmit CommentnPost your commentnnWe inspire you to register/login and move forward.nHowever, if you select to remain non-loggedin, you are required to fill up the sort under and verify your e-mail deal with before we can publish your comment.nThat way, we can confirm that the e-mail deal with doesn't belong to someone else.nRegister/LoginnProceed With out RegistrationnnFull Name nnLocation nnEmail nnFix six + =2nnnnProceed nnPost your commentnnWe inspire you to sign-up/login and move forward.nHowever, if you decide on to continue to be non-loggedin, you are essential to fill up the kind underneath and verify your e-mail address ahead of we can publish your remark.nThat way, we can verify that the electronic mail tackle does not belong to a person else.nRegister/LoginnProceed With out RegistrationnnFull Title nnLocation nnEmail nnFix six + =twonnnnProceed nnnnnThanks! nnWe have despatched you a verification e mail.nTo confirm, just follow the hyperlink in the message nnResend verification mailEmail despatchednnnSort by:Newest|Oldest|Recommended|Most Mentioned|Agree|DisagreeConnect with usFEATURED Today IN US & Canada NewsIndian activist poised to be initial at any time elected socialist in USnIndian-American surgeon gets optimum US immigrant awardnHave world's best task, says US lawful star Preet BhararanPromoting Punjabi heritage in CanadanChhath Puja celebrated in USnmoreYOUR FRIENDS' ACTIVITY1of1FEATURED Today IN NRIIndian jailed in Moscow for $18 million fraudn3 of Indian-origin among Africa's fifty richest: ForbesnIndian origin sisters sentenced for theftnProtests in Uk in help of Indian demise row convict in UAEnIndo-Canadian lady alleges racism in Vancouver barn10 Indians arrested in Uk for immigration offencesn80-year-previous Sikh attacked by lady in British isles diesnRs 1 crore relief for kin of NRI killed in mishapn1 in 10 medical doctors practising in United kingdom is from IndianDo Indians turn out to be far more religious when overseas?nmoreTOP STORIESOnion rates up 278%, tomato 122% in a single yearnAt nine.30am these days, India will appear to a halt as Sachin resumes his innings at 38 in final TestnMeet the thirteen-yr-old lady from Lucknow who is pursuing MSc in microbiologynIndian poised to be very first elected socialist in USnVolvo hearth: Techie smashed pane with head, saved manynmoreMOST POPULARReadn|nCommentedn|nSharednFrom NRI3 of Indian-origin between Africa's 50 richest: ForbesnIndian activist poised to be very first at any time elected socialist in USnIndian-American surgeon gets greatest US immigrant awardnIndian origin sisters sentenced for theftnmoreAcross Moments of IndiaDoodle 4 Google winner's entry marks Children's DaynInd vs WI: Sachin Tendulkar takes India to 157/two at stumps on Day 1 of 2nd TestnRajiv Gandhi wished Bofors cash to run Congress: Ex-CBI chiefn'With Sachin Paaji, a portion of me will be long gone too'nmoreAllnTOInMy ProfilenSign into earn your BadgesKnow far more about Moments PointsTHE Instances OF INDIAnPowered by INDIATIMESnAbout usnnPromote with usnnTerms of Use and Grievance Redressal CoveragennPrivacy plannnFeedbacknnRSSnnNewsletternnTOI MobilennePapernnSitemapnnArchivesnOther Moments Team news sitesnTimes Crest | The Economic Timesn&#x907&#x915&#x928&#x949&#x92e&#x93f&#x915 &#x91f&#x93e&#x907&#x92e&#x94d&#x938 | &#xa88&#xa95&#xacb&#xaa8&#xacb&#xaae&#xabf&#xa95 &#xa9f&#xabe&#xa88&#xaae&#xacd&#xab8nPune Mirror | Bangalore MirrornAhmedabad Mirror | ItsMyAscentnEducation Occasions | Brand CapitalnMumbai Mirror | Times NownIndiatimes | &#x928&#x935&#x92d&#x93e&#x930&#x924 &#x91f&#x93e&#x907&#x92e&#x94d&#x938n&#x92e&#x939&#x93e&#x930&#x93e&#x937&#x94d&#x91f&#x94d&#x930 &#x91f&#x93e&#x907&#x92e&#x94d&#x938 | &#xcb5&#xcbf&#xc9c&#xcaf &#xc95&#xcb0&#xccd&#xca8&#xcbe&#xc9f&#xc95nGo GreenLiving and entertainmentnTimescity | iDiva | Bollywood | ZoomnHealthmeup | Luxpresso | TechnoholiknGuylife | On the internet SongsnnCuriosity Networknitimes | EmailHot on the WebnHotklix | World | PoliticsnBusiness | SportsnEntertainmentnScience & TechnNew CarsServicesnBook print ads | On-line purchasing | Cost-free SMS | Site design | CRM | TendersnMatrimonial | Ringtones | Astrology | Work | Tech Neighborhood | Residence | Acquire carnBikes in India | Offers | Cost-free Classifieds | Send out income to India | Utilised CarsnRestaurants in Delhi | Motion picture Display Timings in Mumbai | Remit to India | Buy MobilesnListen Tracks | Information | Real Estate DevelopersnnTrending TopicsnYahoo | Intercourse | You Tube | Gold Price tag | Twitter | Irctc | Renault Duster | View Videos | Aadhar Card | Facebook | UID CardCopyright � 2013 Bennett, Coleman & Co. Ltd. All legal rights reserved. For reprint legal rights: Moments Syndication ServiceFind this comment offensive?nChoose your reason beneath and click uggs on sale the Submit button. This will notify our moderators to consider actionnNamennExplanation for reporting:nFoul languageSlanderousInciting hatred in opposition to a particular communitySpam/out of contextOthersnnYour Explanation has been Submitted to the admin.nnWhile observing her off to Canada, minor did her neighbours think about that the girl-up coming-doorway in Chandigarh, Guddu, would turn into RJ Aman, and make an identification as a woman aggressively operating to encourage Punjabi heritage amid NRIs settled in Canad

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