Homicidal Necrophobe on a Rampage

I went to AMATYC focused on two things: getting ideas for teaching online this spring and finding good problems. I found plenty of help with both. I’ve posted a lot on online education this week. I have yet to get started with the problems people gave me. Prepare yourself.

Here is the setup for the first: You have a homicidal necrophobe locked up in a jail. The jail is four cells by four cells and all the other cells are filled. There is a key that opens every door, and one night when the jailor goes away for an hour, he comes back and finds all the doors open and everyone dead. Pretty gruesome. Here’s a picture where the X’s represent dead bodies.
The homicidal necrophobe was originally in the first cell that is now empty and is now in the last cell that has two bodies. In the hour the jailor was gone, the HN has killed everybody and then being unable to escape the dead bodies, got so scared he killed himself. The question is what path did the HN take through the jail? Note that he can’t move diagonally only horizontally and vertically, and each time he kills a prisoner he cannot reenter the room after he’s left.

I was given this problem by a sharp young teacher from NYC. Unfortnately, I've lost the notebook with his name and email. If you happen to be that sharp young teacher, I'd love to give you credit. Email me. Thanks for the great problem!