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jbranchaud
43 lines
1.6 KiB
Markdown
43 lines
1.6 KiB
Markdown
# Generate Permutations Of All Valid 9-ball Racks
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I wanted to produce a full listing of all valid rack arrangements for the game
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of [9-ball](https://en.wikipedia.org/wiki/Nine-ball). The constraints on how a
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9-ball rack can be arranged are, first, that the 1 ball must be placed at the
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head of the diamond and, second, that the 9 ball must be placed at the center of
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the diamond. After that, all other balls (2 through 8) can be placed in any
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arrangement.
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Because each of those seven remaining balls can be arranged in distinct
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orderings where each ball is placed once, this is a
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[_permutation_](https://en.wikipedia.org/wiki/Permutation) problem.
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> In elementary combinatorics, the k-permutations, or partial permutations, are
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> the ordered arrangements of k distinct elements selected from a set. When k is
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> equal to the size of the set, these are the permutations in the previous
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> sense.
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For this problem, the seven distinct elements can be arranged into `7!` (seven
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factorial) unique permutations. That is, 5040 permutations.
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I can use [Ruby's `Array#permutations`
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method](https://docs.ruby-lang.org/en/4.0/Array.html#method-i-permutation) to
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enumerate these 5040 permutations like so:
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```ruby
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[2,3,4,5,6,7,8].permutation.map do |perm|
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[1, *perm[0..2], 9, *perm[3..7]]
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end.to_a
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=> [[1, 2, 3, 4, 9, 5, 6, 7, 8],
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[1, 2, 3, 4, 9, 5, 6, 8, 7],
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[1, 2, 3, 4, 9, 5, 7, 6, 8],
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[1, 2, 3, 4, 9, 5, 7, 8, 6],
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[1, 2, 3, 4, 9, 5, 8, 6, 7],
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[1, 2, 3, 4, 9, 5, 8, 7, 6],
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[1, 2, 3, 4, 9, 6, 5, 7, 8],
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...
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[1, 8, 7, 6, 9, 5, 3, 2, 4],
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[1, 8, 7, 6, 9, 5, 3, 4, 2],
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[1, 8, 7, 6, 9, 5, 4, 2, 3],
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[1, 8, 7, 6, 9, 5, 4, 3, 2]]
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```
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