Tumbling Dice by Robert AbbottDate: 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/ |
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