Add Generate Permutations Of All Valid 9-ball Racks as a Math TIL

math/generate-permutations-of-all-valid-9-ball-racks.md

@@ -0,0 +1,42 @@
+# Generate Permutations Of All Valid 9-ball Racks
+
+I wanted to produce a full listing of all valid rack arrangements for the game
+of [9-ball](https://en.wikipedia.org/wiki/Nine-ball). The constraints on how a
+9-ball rack can be arranged are, first, that the 1 ball must be placed at the
+head of the diamond and, second, that the 9 ball must be placed at the center of
+the diamond. After that, all other balls (2 through 8) can be placed in any
+arrangement.
+
+Because each of those seven remaining balls can be arranged in distinct
+orderings where each ball is placed once, this is a
+[_permutation_](https://en.wikipedia.org/wiki/Permutation) problem.
+
+> In elementary combinatorics, the k-permutations, or partial permutations, are
+> the ordered arrangements of k distinct elements selected from a set. When k is
+> equal to the size of the set, these are the permutations in the previous
+> sense.
+
+For this problem, the seven distinct elements can be arranged into `7!` (seven
+factorial) unique permutations. That is, 5040 permutations.
+
+I can use [Ruby's `Array#permutations`
+method](https://docs.ruby-lang.org/en/4.0/Array.html#method-i-permutation) to
+enumerate these 5040 permutations like so:
+
+```ruby
+[2,3,4,5,6,7,8].permutation.map do |perm|
+  [1, *perm[0..2], 9, *perm[3..7]]
+end.to_a
+=> [[1, 2, 3, 4, 9, 5, 6, 7, 8],
+ [1, 2, 3, 4, 9, 5, 6, 8, 7],
+ [1, 2, 3, 4, 9, 5, 7, 6, 8],
+ [1, 2, 3, 4, 9, 5, 7, 8, 6],
+ [1, 2, 3, 4, 9, 5, 8, 6, 7],
+ [1, 2, 3, 4, 9, 5, 8, 7, 6],
+ [1, 2, 3, 4, 9, 6, 5, 7, 8],
+ ...
+ [1, 8, 7, 6, 9, 5, 3, 2, 4],
+ [1, 8, 7, 6, 9, 5, 3, 4, 2],
+ [1, 8, 7, 6, 9, 5, 4, 2, 3],
+ [1, 8, 7, 6, 9, 5, 4, 3, 2]]
+```