There is only one way (up to rotation and reflection) to divide a square into two similar rectangles.
However, there are three distinct ways of partitioning a square into three similar rectangles:[1][2]
The trivial solution given by three congruent rectangles with aspect ratio 3:1.
The solution in which two of the three rectangles are congruent and the third one has twice the side length of the other two, where the rectangles have aspect ratio 3:2.
The solution in which the three rectangles are all of different sizes and where they have aspect ratio ρ2, where ρ is the plastic ratio.
The fact that a rectangle of aspect ratio ρ2 can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ2 related to the Routh–Hurwitz theorem: all of its conjugates have positive real part.[3][4]
In 2022, the mathematician John Baez brought the problem of generalizing this problem to n rectangles to the attention of the Mathstodon online mathematics community.[5][6]
The problem has two parts: what aspect ratios are possible, and how many different solutions are there for a given n.[7] Frieling and Rinne had previously published a result in 1994 that states that the aspect ratio of rectangles in these dissections must be an algebraic number and that each of its conjugates must have a positive real part.[3] However, their proof was not a constructive proof.
Numerous participants have attacked the problem of finding individual dissections using exhaustive computer search of possible solutions. One approach is to exhaustively enumerate possible coarse-grained placements of rectangles, then convert these to candidate topologies of connected rectangles. Given the topology of a potential solution, the determination of the rectangle's aspect ratio can then trivially be expressed as a set of simultaneous equations, thus either determining the solution exactly, or eliminating it from possibility.[8]
The numbers of distinct valid dissections for different values of n, for n = 1, 2, 3, ..., are:[7][9]
1, 1, 3, 11, 51, 245, 1372, 8522, ... (sequence A359146 in the OEIS).
^ abFreiling, C.; Rinne, D. (1994), "Tiling a square with similar rectangles", Mathematical Research Letters, 1 (5): 547–558, doi:10.4310/MRL.1994.v1.n5.a3, MR1295549
