Squaring the square is the problem of

tiling Tiling may refer to: *The physical act of laying tiles * Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester T ...

an integral square using only other integral squares. (An integral square is a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

whose sides have

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

length.) The name was coined in a humorous analogy with

squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficul ...

. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be perfect, meaning the sizes of the smaller squares are all different. A related problem is squaring the plane, which can be done even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a squared square is its number of constituent squares.

Perfect squared squares

A "perfect" squared square is a square such that each of the smaller squares has a different size. It is first recorded as being studied by R. L. Brooks, C. A. B. Smith,

A. H. Stone Arthur Harold Stone (30 September 1916 – 6 August 2000) was a British mathematician born in London, who worked at the universities of Manchester and Rochester, mostly in topology. His wife was American mathematician Dorothy Maharam. Stone s ...

and W. T. Tutte at Cambridge University between 1936 and 1938. They transformed the square tiling into an equivalent

electrical circuit An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage source ...

– they called it a "Smith diagram" – by considering the squares as

resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...

s that connected to their neighbors at their top and bottom edges, and then applied

Kirchhoff's circuit laws Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirc ...

and circuit decomposition techniques to that circuit. The first perfect squared squares they found were of order 69. The first perfect squared square to be published, a compound one of side 4205 and order 55, was found by

Roland Sprague Roland Percival Sprague (11 July 1894, Unterliederbach – 1 August 1967) was a German mathematician, known for the Sprague–Grundy theorem and for being the first mathematician to find a perfect squared square. Biography With two mathematicians ...

in 1939.

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...

published an extensive article written by W. T. Tutte about the early history of squaring the square in his ''Mathematical Games'' column in November 1958. smallest_perfect_squared_squares

Simple squared squares

A "simple" squared square is one where no subset of more than one of the squares forms a rectangle or square, otherwise it is "compound". In 1978, discovered a simple perfect squared square of side 112 with the smallest number of squares using a computer search. His tiling uses 21 squares, and has been proved to be minimal. This squared square forms the logo of the Trinity Mathematical Society. It also appears on the cover of the Journal of Combinatorial Theory. Duijvestijn also found two simple perfect squared squares of sides 110 but each comprising 22 squares. Theophilus Harding Willcocks, an amateur mathematician and fairy chess composer, found another. In 1999, I. Gambini proved that these three are the smallest perfect squared squares in terms of side length. The perfect compound squared square with the fewest squares was discovered by T.H. Willcocks in 1946 and has 24 squares; however, it was not until 1982 that Duijvestijn,

Pasquale Joseph Federico Pasquale ("Pat") Joseph Federico (March 25, 1902 – January 2, 1982) was a lifelong mathematician and longtime high-ranking official of the United States Patent Office. Biography He was born in Monessen, Pennsylvania. About 1910 the family moved t ...

and P. Leeuw mathematically proved it to be the lowest-order example.

Mrs. Perkins's quilt

When the constraint of all the squares being different sizes is relaxed, a squared square such that the side lengths of the smaller squares do not have a common divisor larger than 1 is called a "Mrs. Perkins's quilt". In other words, the

greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

of all the smaller side lengths should be 1. The Mrs. Perkins's quilt problem asks for a Mrs. Perkins's quilt with the fewest pieces for a given

n\times n

square. The number of pieces required is at least

\log_2 n

, and at most

6\log_2 n

. Computer searches have found exact solutions for small values of

n

(small enough to need up to 18 pieces). For

n=1,2,3,\dots

the number of pieces required is:

No more than two different sizes

For any integer

n

other than 2, 3, and 5, it is possible to dissect a square into

n

squares of one or two different sizes.

Squaring the plane

In 1975, Solomon Golomb raised the question whether the whole plane can be tiled by squares, one of each integer edge-length, which he called the heterogeneous tiling conjecture. This problem was later publicized by Martin Gardner in his

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...

column and appeared in several books, but it defied solution for over 30 years. In ''Tilings and Patterns'', published in 1987,

Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentgrew exponentially. For example, the plane can be tiled with different integral squares, but not for every integer, by recursively taking any perfect squared square and enlarging it so that the formerly smallest tile now has the size of the original squared square, then replacing this tile with a copy of the original squared square. In 2008 James Henle and Frederick Henle proved that this, in fact, can be done. Their proof is constructive and proceeds by "puffing up" an L-shaped region formed by two side-by-side and horizontally flush squares of different sizes to a perfect tiling of a larger rectangular region, then adjoining the square of the smallest size not yet used to get another, larger L-shaped region. The squares added during the puffing up procedure have sizes that have not yet appeared in the construction and the procedure is set up so that the resulting rectangular regions are expanding in all four directions, which leads to a tiling of the whole plane.

Cubing the cube

Cubing the cube is the analogue in three dimensions of squaring the square: that is, given a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

''C'', the problem of dividing it into finitely many smaller cubes, no two congruent. Unlike the case of squaring the square, a hard yet solvable problem, there is no perfect cubed cube and, more generally, no dissection of a

rectangular cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...

''C'' into a finite number of unequal cubes. To prove this, we start with the following claim: for any perfect dissection of a ''rectangle'' in squares, the smallest square in this dissection does not lie on an edge of the rectangle. Indeed, each corner square has a smaller adjacent edge square, and the smallest edge square is adjacent to smaller squares not on the edge. Now suppose that there is a perfect dissection of a rectangular cuboid in cubes. Make a face of ''C'' its horizontal base. The base is divided into a perfect squared rectangle ''R'' by the cubes which rest on it. The smallest square ''s''₁ in ''R'' is surrounded by ''larger'', and therefore ''higher'', cubes. Hence the upper face of the cube on ''s''₁ is divided into a perfect squared square by the cubes which rest on it. Let ''s''₂ be the smallest square in this dissection. By the claim above, this is surrounded on all 4 sides by squares which are larger than ''s''₂ and therefore higher. The sequence of squares ''s''₁, ''s''₂, ... is infinite and the corresponding cubes are infinite in number. This contradicts our original supposition. If a 4-dimensional

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...

could be perfectly hypercubed then its 'faces' would be perfect cubed cubes; this is impossible. Similarly, there is no solution for all cubes of higher dimensions.

References

External links

*Perfect squared squares: **, Eindhoven University of Technology, Faculty of Mathematics and Computing Science **http://www.squaring.net/ **http://www.maa.org/editorial/mathgames/mathgames_12_01_03.html **http://www.math.uwaterloo.ca/navigation/ideas/articles/honsberger2/index.shtml **https://web.archive.org/web/20030419012114/http://www.math.niu.edu/~rusin/known-math/98/square_dissect *Nowhere-neat squared squares: **http://karlscherer.com/ *Mrs. Perkins's quilt:
Mrs. Perkins's Quilt
on

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{{Tessellation Discrete geometry Mathematical problems Recreational mathematics Tessellation Geometric dissection