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Undergraduate
By
Shiva Kintali
on Sept. 27, 2013 | Updated Jan. 4, 2018
Fibonacci numbers and Induction
The Fibonacci numbers, \(F_0, F_1, F_2, \dots\) , are defined recursively by the equations \(F_0 = 0\), \(F_1 = 1\), and \(F_n = F_{n-1} + F_{n-2},\) for \(n > 1\). Prove that …
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Undergraduate
By
Shiva Kintali
on June 16, 2012 | Updated Dec. 6, 2017
L shaped tiling
Consider a \(2^n × 2^n\) board with one (arbitrarily chosen) square removed, as in the following figure for \(n = 3\). Prove that any such board can be tiled (without gaps or overlaps) by \(L\)-shaped…
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High School
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Cutting Corners
Prove that removing opposite corner squares from an \(8 \times 8\) chessboard leaves a subboard that cannot be partitioned (tiled) into \(1 \times 2\) and \(2 \times 1\) rectangles. Prove the ab…
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