The Jailer Problem

The Jailer Problem © 1983 Fabris, Robert.

Based on 'The Jailer Problem', a puzzle posed by 'Popular Computing' in the June 1983 issue. Here's the magazine's description:

To recap from last month: an eccentric jailer makes a complicated amnesty offer to his 100 inmates. Every second, he reverses the status of certain cell doors from open to closed or vice versa. After one second, he switches cells whose numbers are multiples of 1 (all of them are opened). After two seconds, cells whose numbers are multiples of 2 (even-numbered cells) are switched. After three seconds, he switches cells whose numbers are multiples of 3, and so on. He repeats this process for 100 seconds. If a cell door is open after the last switch, the con in that cell goes free.