Squaring the square
Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be perfect, meaning that 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.
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 and W. T. Tutte at Cambridge University. They transformed the square tiling into an equivalent electrical circuit — they called it "Smith diagram" — by considering the squares as resistors that connected to their neighbors at their top and bottom edges, and then applied Kirchhoff's circuit laws and circuit decomposition techniques to that circuit.
The first perfect squared square was found by Roland Sprague in 1939.
If we take such a tiling and enlarge it so that the formerly smallest tile now has the size of the square S we started out from, then we see that we obtain from this a tiling of the plane with integral squares, each having a different size.
Simple squared squares 
A "simple" squared square is one where no subset of the squares forms a rectangle or square, otherwise it is "compound". The smallest simple perfect squared square was discovered by A. J. W. Duijvestijn using a computer search. His tiling uses 21 squares, and has been proved to be minimal. The smallest perfect compound squared square was discovered by T.H. Willcocks in 1946 and has 24 squares; however, it was not until 1982 that Duijvestijn, Pasquale Joseph Federico and P. Leeuw mathematically proved it to be the lowest-order example.
The smallest simple squared square forms the logo of the Trinity Mathematical Society.
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 of all the smaller side lengths should be 1.
The Mrs. Perkins's quilt problem is to find a Mrs. Perkins's quilt with the fewest pieces for a given n × n square.
No more than two different sizes 
If a square has to be cut into n squares of at the most two different sizes, without other restrictions, then such positive integer n that it is possible is referred to as a cute number. The Universal Book of Mathematics provides the following information about cute numbers:
- A number n such that a square can be cut into n squares of, at most, two different sizes. For example, 4 and 10 are cute numbers.
It can be shown that, in addition to 1 and 4, every integer greater than 5 is a cute number.
Squaring the plane 
In 1975, Solomon Golomb raised the question whether the whole plane can be tiled by squares whose sizes are all natural numbers without repetitions, which he called the heterogeneous tiling conjecture. This problem was later publicized by Martin Gardner in his Scientific American column and appeared in several books, but it defied solution for over 30 years. In Tilings and Patterns, published in 1987, Branko Grünbaum and G. C. Shephard stated that in all perfect integral tilings of the plane known at that time, the sizes of the squares grew exponentially.
Recently, 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 C, the problem of dividing it into finitely many smaller cubes, no two congruent.
Unlike the case of squaring the square, a hard but solvable problem, cubing the cube is impossible. This can be shown by a relatively simple argument. Consider a hypothetical cubed cube. The bottom face of this cube is a squared square; lift off the rest of the cube, so you have a square region of the plane covered with a collection of cubes.
Consider the smallest cube in this collection, with side c (call it S). Since the smallest square of a squared square cannot be on its edge, its neighbours will all tower over it, meaning that there isn't space to put a cube of side larger than c on top of it. Since the construction is a cubed cube, you're not allowed to use a cube of side equal to c; so only smaller cubes may stand upon S. This means that the top face of S must be a squared square, and the argument continues by infinite descent. Thus it is not possible to dissect a cube into finitely many smaller cubes of different sizes.
Similarly, it is impossible to hypercube a hypercube, because each cell of the hypercube would need to be a cubed cube, and so on into the higher dimensions.
- "Compound Perfect Squares", By A. J. W. Duijvestijn, P. J. Federico, and P. Leeuw, Published in American Mathematical Monthly Volume 89 (1982) pp 15-32
- The Encyclopedia of Science - David Darling
- Henry, JB; Taylor, PJ. Challenge! 1999 - 2006 Book 2. Australian Mathematics Trust. p. 84. ISBN 978-1-876420-23-9.
- C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of Orders 21 Through 25, Eindhoven Univ. Technology, Dept. of Math., Report 92-WSK-03, Nov. 1992.
- C. J. Bouwkamp and A. J. W. Duijvestijn, Album of Simple Perfect Squared Squares of order 26, Eindhoven University of Technology, Faculty of Mathematics and Computing Science, EUT Report 94-WSK-02, December 1994.
- Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; and Tutte, W. T. The Dissection of Rectangles into Squares, Duke Math. J. 7, 312–340, 1940
- Martin Gardner, "Squaring the square," in The 2nd Scientific American Book of Mathematical Puzzles and Diversions.
- Frederick V. Henle and James M. Henle, "Squaring the plane", American Mathematical Monthly, 115, January 2008, 3–12
- Perfect squared squares:
- Nowhere-neat squared squares: