Rectangle Tiling
RectangleTiling
The number of ways N(m,n) of finding a subrectangle with an m×n rectangle can be computed by counting the number of ways in which the upper right-hand corner can be selected for a given lower left-hand corner. For a lower left-hand corner with coordinates (i,j), there are (m-i)(n-j) possible upper right-hand corners, so
N(m,n) = [画像:sum_(i=0)^(m-1)sum_(j=0)^(n-1)(m-i)(n-j)]
(1)
= 1/4m(m+1)n(n+1).
(2)
Equivalently, N(m,n) is the number of ways of picking two lines out of sets of m+1 and n+1 lines, giving
N(m,n) = [画像:(m+1; 2)(n+1; 2)]
(3)
= 1/4m(m+1)n(n+1),
(4)
as before. Particular tilings are shown above for 2×2 and 2×3 rectangles.
See also
Perfect Rectangle, Rectangle, Triangle TilingExplore with Wolfram|Alpha
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References
Stewart, I. "Squaring the Square." Sci. Amer. 277, 94-96, July 1997.Referenced on Wolfram|Alpha
Rectangle TilingCite this as:
Weisstein, Eric W. "Rectangle Tiling." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RectangleTiling.html