Square

[[File:Plastic square partitions.svg|thumb|Partitions of a square into three similar rectangles]]

[[File:Plastic square partitions.svg|thumb|Partitions of a square into three similar rectangles]]

Another counting problem involving squares asks for the number of different shapes of rectangle that can be used when [[dividing a square into similar rectangles]].<ref>{{cite news |last=Roberts |first=Siobhan | author-link = Siobhan Roberts |date=February 7, 2023 |title=The quest to find rectangles in a square|newspaper=[[The New York Times]] |url=https://www.nytimes.com/2023/02/07/science/puzzles-rectangles-mathematics.html}}</ref> A square can be divided into two similar rectangles only in one way, by bisecting it, but when dividing a square into three similar rectangles there are three possible [[aspect ratio]]s of the rectangles, 3:1, 3:2, and the square of the [[plastic ratio]]. The number of proportions that are possible when dividing into <math>n</math> rectangles is known for small values of <math>n</math>, but not as a general formula. For <math>n=1,2,3,\dots</math> these numbers are:<ref>{{cite OEIS|A359146|Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible}}</ref>

Another counting problem involving squares asks for the number of different shapes of rectangle that can be used when [[dividing a square into similar rectangles]].<ref>{{cite news |last=Roberts |first=Siobhan | author-link = Siobhan Roberts |date=February 7, 2023 |title=The quest to find rectangles in a square|newspaper=[[The New York Times]] |url=https://www.nytimes.com/2023/02/07/science/puzzles-rectangles-mathematics.html}}</ref> A square can be divided into two similar rectangles only in one way, by bisecting it, but when dividing a square into three similar rectangles there are three possible [[aspect ratio]]s of the rectangles, 3:1, 3:2, and the square of the [[plastic ratio]], approximately 1.755:1. The number of proportions that are possible when dividing into <math>n</math> rectangles is known for small values of <math>n</math>, but not as a general formula. For <math>n=1,2,3,\dots</math> these numbers are:<ref>{{cite OEIS|A359146|Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible}}</ref>

{{block indent|left=1.6|1, 1, 3, 11, 51, 245, 1372, ...}}

{{block indent|left=1.6|1, 1, 3, 11, 51, 245, 1372, ...}}