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| [[File:Prince Ruperts cube.png|thumb|A unit cube with a hole cut through it, large enough to allow Prince Rupert's cube to pass]]
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| In [[geometry]], '''Prince Rupert's cube''' (named after [[Prince Rupert of the Rhine]]) is the largest [[cube]] that can pass through a hole cut through a unit [[cube]], i.e. through a cube whose sides have length 1. Its side length is approximately 6% larger than that of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution.<ref name="rickey"/><ref name="jw04"/><ref name="gardner"/>
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| ==Solution==
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| If two points are placed on two adjacent edges of a unit cube, each at a distance of 3/4 from the point where the two edges meet, then the distance between the two points will be
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| : <math>\frac{3\sqrt{2}}{4} \approx 1.0606601.</math>
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| These two points, together with a second set of two points placed symmetrically on the opposite face of the cube, form the four vertices of a square that lies entirely within the unit cube. This square, extruded in both directions perpendicularly to itself, forms the hole through which a cube larger than the original one (up to side length <math>\tfrac{3\sqrt{2}}{4}</math>) may pass.<ref name="gardner"/>
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| The parts of the unit cube that remain, after emptying this hole, form two [[triangular prism]]s and two irregular [[tetrahedron|tetrahedra]], connected by thin bridges at the four vertices of the square. | |
| Each prism has as its six vertices two adjacent vertices of the cube, and four points along the edges of the cube at distance 1/4 from these cube vertices. Each tetrahedron has as its four vertices one vertex of the cube, two points at distance 3/4 from it on two of the adjacent edges, and one point at distance 3/16 from the cube vertex along the third adjacent edge.<ref name="wells"/>
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| ==History==
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| Prince Rupert's cube is named after [[Prince Rupert of the Rhine]]. According to a story recounted in 1693 by English mathematician [[John Wallis]], Prince Rupert wagered that a hole could be cut through a cube, large enough to let another cube of the same size pass through it. Wallis showed that in fact such a hole was possible (with some errors that were not corrected until much later), and Prince Rupert won his wager.<ref name="rickey"/><ref name="jw04"/>
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| Wallis assumed that the hole would be parallel to a [[space diagonal]] of the cube. The [[Orthographic projection|projection]] of the cube onto a plane perpendicular to this diagonal is a [[regular hexagon]], and the best hole parallel to the diagonal can be found by drawing the largest possible square that can be inscribed into this hexagon. Calculating the size of this square shows that a cube with side length
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| :<math>\sqrt 6 -\sqrt 2\approx 1.03527</math>,
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| slightly larger than one, is capable of passing through the hole.<ref name="rickey"/>
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| Approximately 100 years later, Dutch mathematician [[Pieter Nieuwland]] found that a better solution (in fact, the optimal solution) may be achieved by using a hole with a different angle than the space diagonal. Nieuwland died in 1794 (a year after taking a position as a professor at the [[University of Leiden]]) but his solution was published posthumously in 1816 by Nieuwland's mentor, [[Jean Henri van Swinden]].<ref name="rickey">{{citation|url=http://www.math.usma.edu/people/Rickey/papers/ShortCourseAlbuquerque.pdf|title=Dürer’s Magic Square, Cardano’s Rings, Prince Rupert’s Cube, and Other Neat Things|year=2005|first=V. Frederick|last=Rickey}}. Notes for “Recreational Mathematics: A Short Course in Honor of the 300th Birthday of Benjamin Franklin,” Mathematical Association of America, Albuquerque, NM, August 2-3, 2005.</ref><ref name="jw04">{{citation
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| | last1 = Jerrard | first1 = Richard P.
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| | last2 = Wetzel | first2 = John E.
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| | doi = 10.2307/4145012
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| | issue = 1
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| | journal = The American Mathematical Monthly
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| | mr = 2026310
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| | pages = 22–31
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| | title = Prince Rupert's rectangles
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| | volume = 111
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| | year = 2004}}.</ref>
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| Since then, the problem has been repeated in many books on [[recreational mathematics]], in some cases with Wallis' suboptimal solution instead of the optimal solution.<ref name="gardner"/><ref name="wells">{{citation|title=The Penguin Dictionary of Curious and Interesting Numbers|first=David|last=Wells|edition=3rd|publisher=Penguin|year=1997|isbn=9780140261493|url=http://books.google.com/books?id=kQRPkTkk_VIC&pg=PA16|page=16}}.</ref><ref>{{citation|title=Recreations in Mathematics and Natural Philosophy: Containing Amusing Dissertations and Enquiries Concerning a Variety of Subjects the Most Remarkable and Proper to Excite Curiosity and Attention to the Whole Range of the Mathematical and Philosophical Sciences|first=Jacques|last=Ozanam|authorlink=Jacques Ozanam|editor1-first=Jean Étienne|editor1-last=Montucla|editor1-link=Jean-Étienne Montucla|editor2-first=Charles|editor2-last=Hutton|editor2-link=Charles Hutton|publisher=G. Kearsley|year=1803|pages=315–316|url=http://books.google.com/books?id=s_IJAAAAMAAJ&pg=PA315}}.</ref><ref>{{citation|title=Modern puzzles and how to solve them|first=Henry Ernest|last=Dudeney|authorlink=Henry Dudeney|year=1936|page=149}}</ref><ref>{{citation
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| | last = Ogilvy | first = C. Stanley
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| | pages = 54–55
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| | publisher = Oxford University Press
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| | title = Through the Mathescope
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| | year = 1956}}. Reprinted as {{citation
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| | last = Ogilvy | first = C. Stanley
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| | isbn = 0-486-28283-X
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| | location = New York
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| | mr = 1313725
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| | publisher = Dover Publications Inc.
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| | title = Excursions in mathematics
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| | url = http://books.google.com/books?id=WLcTi34V1ecC&pg=PA54
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| | year = 1994}}.</ref><ref>{{citation | |
| | last = Ehrenfeucht | first = Aniela
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| | location = New York
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| | mr = 0170242
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| | page = 77
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| | publisher = The Macmillan Co.
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| | title = The cube made interesting
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| | year = 1964}}. Translated from the Polish by Waclaw Zawadowski.</ref><ref>{{citation|title=[[Flatterland|Flatterland: Like Flatland Only More So]]|first=Ian|last=Stewart|authorlink=Ian Stewart (mathematician)|publisher=Macmillan|year=2001|isbn=9780333783122|pages=49–50}}.</ref><ref>{{citation|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|first=David|last=Darling|publisher=John Wiley & Sons|year=2004|isbn=9780471667001|url=http://books.google.com/books?id=HrOxRdtYYaMC&pg=PA255|page=255}}.</ref><ref>{{citation|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics|first=Clifford A.|last=Pickover|authorlink=Clifford Pickover|publisher=Sterling Publishing Company, Inc.|year=2009|isbn=9781402757969|url=http://books.google.com/books?id=JrslMKTgSZwC&pg=PA214|page=214}}.</ref>
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| ==Models==
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| The construction of a physical model of Prince Rupert's cube is made difficult by the accuracy with which such a model would need to be measured, and the thinness of the connections between the remaining parts of the unit cube after the hole is cut through it; for this reason, the problem has been called "mathematically possible but practically impossible".<ref>{{citation|title=Interdisciplinarity, Creativity, and Learning: Mathematics With Literature, Paradoxes, History, Technology, and Modeling|volume=7|series=Montana Mathematics Enthusiast Monograph Series in Mathematics Education|editor1-first=Bharath|editor1-last=Sriraman|editor2-first=Viktor|editor2-last=Freiman|editor3-first=Nicole|editor3-last=Lirette-Pitre|year=2009|isbn=9781607521013|publisher=Information Age Publishing, Inc.|contribution=Mathematics and literature (the sequel): imagination as a pathway to advanced mathematical ideas and philosophy|first=Bharath|last=Sriraman|pages=41–54}}.</ref> Nevertheless, in a 1950 survey of the problem, D. J. E. Schrek published photographs of a model of a cube passing through a hole in another cube.<ref>{{citation|last=Schrek|first= D. J. E.|title=Prince Rupert’s problem and its extension by Pieter Nieuwland|journal=[[Scripta Mathematica]]|volume=16|year=1950|pages=73–80 and 261–267}}. As cited by {{harvtxt|Rickey|2005}} and {{harvtxt|Jerrard|Wetzel|2004}}.</ref>
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| Martin Raynsford has designed a template for constructing paper models of a cube with another cube passing through it; in order to account for the tolerances of paper construction and not tear the paper at the narrow joints between parts of the punctured cube, the hole in Raynsford's model is slightly smaller than the cube it lets pass through.<ref>{{citation|first=George W.|last=Hart|authorlink=George W. Hart|title=Math Monday: Passing a Cube Through Another Cube|publisher=Museum of Mathematics|date=January 30, 2012|url=http://momath.org/home/math-monday-passing-a-cube-through-another-cube/}}. Originally published in ''[[Make (magazine)|Make Online]]''.</ref>
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| ==Generalizations==
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| The cube is not the only body that can pass through a hole cut into a copy of itself; the same is true for the regular [[tetrahedron]] and [[octahedron]].<ref>{{citation
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| | last = Scriba | first = Christoph J.
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| | issue = 9
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| | journal = Praxis der Mathematik
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| | language = German
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| | mr = 0497615
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| | pages = 241–246
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| | title = Das Problem des Prinzen Ruprecht von der Pfalz
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| | volume = 10
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| | year = 1968}}.</ref>
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| Another way to express the same problem is to ask for the largest [[square]] that lies within a unit cube. More generally, {{harvtxt|Jerrard|Wetzel|2004}} show how to find the largest [[rectangle]] of a given [[aspect ratio]] that lies within a unit cube. As they show, the optimal rectangle must always pass through the center of the cube, with its vertices on edges of the cube. Based on this, they show, depending on the desired aspect ratio, that the optimal rectangle must either lie on a plane that cuts diagonally through four corners of the cube, or it must be formed by an isosceles right triangle on one corner of the cube and by the two opposite points, as in the case of Prince Rupert's problem.<ref name="jw04"/> If the aspect ratio is not constrained, the rectangle with the largest area that fits within a cube is the one that has two opposite edges of the cube as two of its sides, and two face diagonals as the other two sides.<ref>{{citation|title=Calculus Made Easy|first1=Silvanus P.|last1=Thompson|first2=Martin|last2=Gardner|author2-link=Martin Gardner|edition=3rd|publisher=Macmillan|year=1998|isbn=9780312185480|url=http://books.google.com/books?id=BBIFtid-WdUC&pg=PA315|page=315}}.</ref>
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| Alternatively, one may ask for the largest <math>m</math>-dimensional hypercube that may be drawn within an <math>n</math>-dimensional unit [[hypercube]]. The answer is always an [[algebraic number]]. For instance, the problem for <math>(m,n)=(3,4)</math> asks for the largest cube within a four-dimensional hypercube. After [[Martin Gardner]] posed this question in ''[[Scientific American]]'', Kay R. Pechenick DeVicci and several other readers showed that the answer for the (3,4) case is the [[square root]] of the smaller of two [[root of a function|real roots]] of the [[polynomial]] <math>4x^4-28x^3-7x^2+16x+16</math>, which works out to approximately 1.007435.<ref name="gardner">{{citation|title=The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics|first=Martin|last=Gardner|authorlink=Martin Gardner|publisher=W. W. Norton & Company|year=2001|isbn=9780393020236|pages=172–173|url=http://books.google.com/books?id=orz0SDEakpYC&pg=PA172}}.</ref><ref>{{citation
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| | last1 = Guy | first1 = Richard K. | author1-link = Richard K. Guy
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| | last2 = Nowakowski | first2 = Richard J.
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| | doi = 10.2307/2974481
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| | issue = 10
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| | journal = [[The American Mathematical Monthly]]
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| | mr = 1543116
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| | pages = 967–973
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| | title = Unsolved Problems: Monthly Unsolved Problems, 1969-1997
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| | volume = 104
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| | year = 1997}}.</ref> For <math>m=2</math>, the optimal side length of the largest square in an <math>n</math>-dimensional hypercube is either <math>\sqrt{n/2}</math> or <math>\sqrt{n/2-3/8}</math>, depending on whether <math>n</math> is even or odd respectively.<ref>{{mathworld | urlname = CubeSquareInscribing | title = Cube Square Inscribing}}</ref>
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *{{mathworld | urlname = PrinceRupertsCube | title = Prince Rupert's Cube}}
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| [[Category:Cubes]]
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