Cube of Crete

Wizard Card  -  Volume 4  -  Mr. Wizard Number 1  -  Mon, Feb 6, 1989 5:19 AM

After many unexpected delays and perilous encounters with defective hardware, THE WIZARD IS BACK! In this installment I take up what is perhaps my favorite subject: mazes! This will be the first of many different maze stacks.

The maze featured in this stack is the creation of A. K. Dewdney and first appeared in the September 1988 issue of Scientific American. The moment I saw it, I fell in love with it: it is a thing of beauty. The maze is a fanciful reconstruction of Daedalus' labyrinth of Crete, the one with the horrible Minotaur at the center.

Dewdney's maze is quite simple except for one thing: it is three dimensional. We usually think of ourselves as three dimensional beings but for all practical purposes we are flatlanders. Even a simple three dimensional maze like this one confuses us because we are not used to moving in three dimensions. If you watch birds or flies closely you will frequently be surprised by the moves they make, the drops and swerves, and somersaults. They move in ways that simply don't occur to us. How limited we must seem to them!

In order to make this particular maze a little easier for you poor flatlanders, I provide a complete floorplan of the maze right from the start. The exit is not marked but you'll know it when you try it.

In order to represent all the shafts and holes I have adopted Dewdney's excellent convention: holes in the floor appear as small dark squares. Holes in the ceiling appear as slightly larger open squares. Occasionally both squares will appear superimposed to indicate a hole in both the floor and the ceiling (usually a mine shaft).

The maze is organized as a six by six by six cube. There are six different levels, level one being the one nearest the surface. To move through a hole to another room, simply click on the appropriate square hole marker. Most of the passages in the maze have only a single entrance and a single exit; your only choice is turn back or keep going.

To speed things up this program automatically keeps going. That is, once you choose a hole, the computer will move you from passage to passage until you come to a room with more than two exits. The computer will then stop and wait for you to click on another hole. Each time you choose an exit, that exit marker will flash quickly on and off as will all the other exits that the computer chooses for you. When you arrive at a decision point, the entrance hole will flash slowly three times. You start on level one.

For your added amusement, I have added another bizarre feature. Each decision point you come to is inhabited by a different animal. You can't see them but you can certainly hear them! Thus this maze is also an auditory maze; you keep moving from sound to sound until you get out.

Oh, here's a riddle for you. This maze is cleverly designed so that the Minotaur can stand on a certain spot and be certain that no matter where you wander you will eventually have to cross his path. Can you identify that spot?

As soon as you escape the maze you will be given a chance to see an instant replay of your path through the maze with a timer that shows exactly how long it took you. You will also be given a chance to view the two dimensional structure of the maze. I believe you will be surprised at just how simple this maze really is. The 2D diagram can be animated to replay your path. You can also click on the various animal symbols and see exactly how each node in the 2D diagram relates to the original maze.

How long will it take? That is entirely up to you. It's possible to get through the maze in less than thirty seconds or you could be in there for ten minutes or more. It's up to you whether you want to spend time studying the floorplan or just plunge in and start jumping through holes.

It IS possible to calculate the average amount of time that a randomly moving mouse would take. This requires a technique called Markov Analysis that I will probably discuss in some future column. At any rate, if the mouse makes one move every ten seconds I calculate that the mouse will emerge in exactly ten minutes on average (although he COULD be in there for centuries). Along the way the mouse would encounter 9.64 cows, 7.73 wolves, 9.64 Sealions, 6.73 Elephants, 7.64 cheshire cats, 4.09 bullfrogs, 5.82 horses, 4 owls, and 4.73 wasps. You should be able to do much better than the mouse.

Push the "Push Me" button to begin. Good luck!