Editing Square (section)

===Counting===
{{main|Square pyramidal number|Dividing a square into similar rectangles}}
[[File:Two square counting puzzles.svg|thumb|upright=1.2|Two square-counting puzzles: There are 14 squares in a {{math|3 × 3}} grid of squares (top), but as a {{math|4 × 4}} grid of points it has six more off-axis squares (bottom) for a total of 20.]]
A common [[mathematical puzzle]] involves counting the squares of all sizes in a square grid of <math>n\times n</math> squares. For instance, a square grid of nine squares has 14 squares: the nine squares that form the grid, four more <math>2\times 2</math> squares, and one <math>3\times 3</math> square. The answer to the puzzle is <math>n(n+1)(2n+1)/6</math>, a [[square pyramidal number]].<ref>{{cite journal
 | last1 = Duffin | first1 = Janet
 | last2 = Patchett | first2 = Mary
 | last3 = Adamson | first3 = Ann
 | last4 = Simmons | first4 = Neil
 | date = November 1984
 | issue = 5
 | journal = Mathematics in School
 | jstor = 30216270
 | pages = 2–4
 | title = Old squares new faces
 | volume = 13}}</ref> For <math>n=1,2,3,\dots</math> these numbers are:<ref>{{cite OEIS|A000330|Square pyramidal numbers}}</ref>

{{block indent|left=1.6|1, 5, 14, 30, 55, 91, 140, 204, 285, ...}}

A variant of the same puzzle asks for the number of squares formed by a grid of <math>n\times n</math> points, allowing squares that are not axis-parallel. For instance, a grid of nine points has five axis-parallel squares as described above, but it also contains one more diagonal square for a total of six.<ref>{{cite journal
 | last = Bright | first = George W.
 | date = May 1978
 | doi = 10.5951/at.25.8.0039
 | issue = 8
 | journal = The Arithmetic Teacher
 | jstor = 41190469
 | pages = 39–43
 | publisher = National Council of Teachers of Mathematics
 | title = Using Tables to Solve Some Geometry Problems
 | volume = 25}}</ref> In this case, the answer is given by the ''4-dimensional pyramidal numbers'' <math>n^2(n^2-1)/12</math>. For <math>n=1,2,3,\dots</math> these numbers are:<ref>{{cite OEIS|A002415|4-dimensional pyramidal numbers}}</ref>

{{block indent|left=1.6|0, 1, 6, 20, 50, 105, 196, 336, 540, ...}}

[[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>

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