rosetta-code.josephus-problem: adding solution for josephus problem.

extra/rosetta-code/josephus-problem/josephus-problem.factor

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+USING: kernel locals math math.ranges sequences ;
+IN: rosetta-code.josephus-problem
+
+! http://rosettacode.org/wiki/Josephus_problem
+
+! Problem: Josephus problem is a math puzzle with a grim
+! description: n prisoners are standing on a circle, sequentially
+! numbered from 0 to n − 1. An executioner walks along the circle,
+! starting from prisoner 0, removing every k-th prisoner and
+! killing him. As the process goes on, the circle becomes smaller
+! and smaller, until only one prisoner remains, who is then freed.
+! For example, if there are n = 5 prisoners and k = 2, the order
+! the prisoners are killed in (let's call it the "killing
+! sequence") will be 1, 3, 0, and 4, and the survivor will be #2.
+
+! Task: Given any n,k > 0, find out which prisoner will be the
+! final survivor. In one such incident, there were 41 prisoners
+! and every 3rd prisoner was being killed (k = 3). Among them was
+! ! a clever chap name Josephus who worked out the problem, stood at
+! the surviving position, and lived on to tell the tale. Which
+! number was he?
+
+! Extra: The captors may be especially kind and let m survivors
+! free, and Josephus might just have m − 1 friends to save.
+! Provide a way to calculate which prisoner is at any given
+! position on the killing sequence.
+
+! Notes:
+! 1. You can always play the executioner and follow the
+!    procedure exactly as described, walking around the circle,
+!    counting (and cutting off) heads along the way. This would yield
+!    the complete killing sequence and answer the above questions,
+!    with a complexity of probably O(kn). However, individually it
+!    takes no more than O(m) to find out which prisoner is the m-th
+!    to die.
+! 2. If it's more convenient, you can number prisoners from 1 to
+!    n instead. If you choose to do so, please state it clearly.
+! 3. An alternative description has the people committing
+!    assisted suicide instead of being executed, and the last person
+!    simply walks away. These details are not relevant, at least not
+!    mathematically.
+
+:: josephus ( n k -- m )
+    n [1,b] 0 [ [ k + ] dip mod ] reduce ;
+
+: josephus2 ( n -- m )  ! faster for k=2
+    dup log2 2^ - 2 * ;