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Number of ways of cutting a rectangle into 1x2 rectangles

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December 6, 2010
geometryrectangleinductioncombinatorics unsolvedcombinatorics

Problem Statement

A rectangle ABCDABCD is given whose sides have lengths 33 and 2n2n, where nn is a natural number. Denote by U(n)U(n) the number of ways in which one can cut the rectangle into rectangles of side lengths 11 and 22. (a)(a) Prove that U(n+1)+U(n−1)=4U(n);U(n + 1)+U(n -1) = 4U(n); (b)(b) Prove that U(n)=123[(3+1)(2+3)n+(3−1)(2−3)n].U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].
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