Tumbling Dice by Robert Abbott

Date: 03/17/2002 at 20:07:02
From: Jennifer Reichel
Subject: Tumbling Dice by Robert Abbott listed as a Brain Boggler

Dear Dr. Math:

Here are the instructions for Tumbling Dice.

4 1 5 6 * 6 4
6 2 4 * 1 1 2
2 3 6 3 2 * *
4 5 5 5 6 2 1
6 * 4 2 3 1 3
4 2 3 1 * 4 *
6 1 5 4 5 6 2
 
Look around the house for a game with dice in it. Remove one die and 
place it on the red 5 in the center of the maze above. Place the die 
so that the 6 is on top and the 4 is facing you.

Now, try thinking of the die as a large carton that is too heavy to 
slide to another square. You can, however, tip it over on an edge 
and have it land on an adjacent square. In that manner you move the 
die horizontally or vertically from one square to another. There is 
also this restriction: you can only move onto a square that contains 
the same number as the number on top of the die. However, a square 
with a star is a "wild" square. You can move to one of these squares 
no matter what number is on top of the die.

Here's an example: When we start the maze, the only possible move is 
to go right, onto the square with the 6. If we tip the die onto that 
square, a 2 will now appear on top. We can now move up to a square 
with a 2 or we can go right to a square with a 2. Suppose we move 
right. A 1 is now on top of the die. At this point we can go right or 
down to a square with a 1 , or we could go up to a wild square. Let's 
say we go down. A 3 is now on top, and we can go right or left. We go 
right and a 5 appears on top. The only possible move is down onto the 
wild square. A 6 now appears on the top. There is no 6 adjacent to us, 
so we have reached a dead end and have to go start all over.

The center square is both the start and the goal: to solve the maze, 
you must move the die off the center square, then find a way to move 
it back onto that square.

Thank you, 
Jennifer

Date: 03/18/2002 at 23:06:47
From: Doctor Peterson
Subject: Re: Tumbling Dice by Robert Abbott listed as a Brain Boggler

Hi, Jennifer.

We prefer to help people solve problems themselves, so they can learn 
from them. In this case, it's hard to give any real guidance, because 
there's really nothing to this except patience and an orderly way of 
keeping track of what you've tried. I finally managed to solve it 
tonight after working at it longer than I want to admit; I'm pretty 
sure I found the only solution, because I had no more possible routes 
left when I found it!

My solution takes 48 steps, going through 35 of the squares on the 
board, seven of them twice and two of them three times. It took some 
time even to find any paths that go over to the left side of the 
board, and it took a bit of a trick to do that, involving a loop that 
you might not think was worth trying. For that reason, I will give you 
the beginning of my path:

    Start at 5
    Go right to 6, then 2, then 1
    Go up to * and left to the next *
    Go up, right, and down through 1 and 2 and back to the first *
    Go left through the second * again and the 2
    Now go down for a while

See if you can find it with that start. It's amazing to see how many 
different paths can be squeezed into the same space by making the 
orientation of the die determine which paths are allowed.

I assume you have an orderly way to record paths you've tried and 
directions you've left to try later. I just drew lines for the paths I 
took, ending with an X when they ended and a dot where I hadn't tried 
continuing one of two or more possibilities from one square; and a had 
a way to record with each dot what position the die was in when I left 
it (what was on top and on the front). That's really all it takes - a 
little insane persistence and a plan.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/