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Discrete Mathematics An Open Introduction, 4th Edition

Oscar Levin

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Let’s find some of the numbers of paths that the rook can take to get to various squares in the chessboard.
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1.
The 6 in the square in the 3rd row and column represents that there are 6 different paths to that square, even though there are only four squares the rook must move through to get there. One path is DDRR (down down right right). List all 6 paths.
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Answer.
\(\text{DRDR}, \text{DRRD}, \text{RDDR}, \text{RRDD}, \text{RDRD}, \text{DDRR}\)
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2.
How many paths are there to the square in row 4, column 2 (diagonally down and to the left of the 6)? List out all the paths as D/R strings.
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How many paths is this? That is, what number goes in that square of the chessboard?
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Answer 1.
\(\text{DDDR}, \text{DDRD}, \text{DRDD}, \text{RDDD}\)
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Answer 2.
\(4\)
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3.
Now let’s find the paths to the square in row 4, column 3 (directly below the 6).
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First, list all the paths that end with an R.
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Next, list all the paths that end with a D.
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Are there any other paths? In total, how many paths are there to this square?
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Answer 1.
\(\text{DDDRR}, \text{DDRDR}, \text{DRDDR}, \text{RDDDR}\)
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Answer 2.
\(\text{DRDRD}, \text{DRRDD}, \text{RDDRD}, \text{RRDDD}, \text{RDRDD}, \text{DDRRD}\)
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Answer 3.
\(10\)
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4.
Continue filling in the chessboard, either counting D/R strings directly or using your observation from the previous task. What is the number in the lower right corner of the chessboard?
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Answer.
\(3432\)
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