I am a high school math teacher in Canada (Pitt Meadows, near Vancouver). Last year I was teaching a grade 9 enriched math course, and I had the students do a 6 week mathematical investigation of a problem I gave them. I had a young Korean girl who solved her first investigation in 2 days (as opposed to 6 weeks), so I gave her a second investigation: the riddle of Josephus. She returned a few days later with a very elegant solution for the case where every other person is killed. I'll tell you how she explained it to me, and then we'll get a little formal as to the proof. She'd only been in Canada for 2 months, so her English wasn't the greatest. First, she wrote how many people were in the circle.
Then she began writing powers of 2: 2,4,8,16,32,64. She stopped, and circled the 32. She then divided the 32 into 50 getting a remainder of 18. She doubled it, getting 36. The 36th person will remain standing. What a wonderfully original, innovative solution! Our class tested it for circles of various sizes, and it works. I haven't sat down and proved it formally (maybe during Christmas), but I wanted to pass this on to you and to other high school teachers. Allow your kids to be able to spend a fair bit of time wrestling with the problem; their solutions may surprise you.