ps5sol - Massachusetts Institute of Technology...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Fall 05 : Mathematics for Computer Science October 28 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised October 12, 2005, 909 minutes Solutions to Problem Set 5 Problem 1. Suppose that one domino can cover exactly two squares on a chessboard, either vertically or horizontally. (a) Can you tile an 8 8 chessboard with 32 dominos? chess board dominos Solution. Yes. Place 4 vertical dominos in each column. (b) Can you tile an 8 8 chessboard with 31 dominos if opposite corners are removed? Solution. No! Opposing corners are the same color. Therefore, removing opposite cor- ners leaves an unequal number of white and black squares. Since every domino covers one black square and one white square, no tiling is possible. Copyright 2005, Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld . All rights reserved. Solutions to Problem Set 5 2 (c) Now suppose that an assortment of squares are removed from a chessboard. An example is shown below. Given a truncated chessboard, show how to construct a bipartite graph G that has a per- fect matching if and only if the chessboard can be tiled with dominos. Solution. Create a vertex for every white square and a vertex for every black square. Put an edge between squares that share an edge. (This graph is bipartite, since the coloring ofan edge between squares that share an edge....
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This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.

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ps5sol - Massachusetts Institute of Technology...

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