Proving Math Logic Problem: Puzzle 8 Configuration

• j9mom
In summary, the task at hand is to prove that the given puzzle configuration cannot be solved by making any number of moves, as is possible in a typical puzzle. The approach suggested is to exhaust all possible moves and their resulting configurations, and prove that the desired configuration cannot be achieved. Another potential approach is to assign a number to each game position and demonstrate that a move can only change this number by two, making it impossible to reach the desired odd or even final number. This problem is similar to the well-known '15 puzzle' and can be researched further using online resources.
j9mom

Homework Statement

I am supposed to prove that if you have a puzzle 8 in this configuration

A B C
D E F
H G

I have to prove that no matter how many moves you make (as you
would a normal puzzle moving one of the adjacent letters into the
blank space) you cannot make the configuration

A B C
D E F
G H

I know you can't. I have moved these letters around all over the place... but how do I start to prove it... I don't know where to start

The Attempt at a Solution

I just want a starting point...

This is an interesting problem.

Isn't the idea that you can slide a letter left or right, or up or down into an empty spot? The physical puzzles I've seen like this won't let you slide a letter diagonally.

I don't know for sure, but I think that what you need to do here is to look at all the ways the blank can move around, starting from the "home" position, and getting back to it.

For starters, the blank could move all around the 2x2 square at the lower right. At each of the four positions what is state of the puzzle.
It could move around the two rightmost columns. At each of the six positions, what is the state of the puzzle?
It could move all around the two lower rows. At each of the six positions, what is the state of the puzzle?
It could move all around the perimeter of the puzzle. At each of the eight positions, what is the state of the puzzle?
I can't think of any other possibilities, although there might be some I haven't counted.

Once you exhaust the possible moves, and the one you're looking for doesn't appear, that would convince me that it's not possible. You have to make sure that you haven't omitted any possible puzzle states, though.

The proof of impossibility for games like this involves assigning a even or odd number to each game position and then showing that a move in the game can only change that number by two. So if the starting position is even and the final position is odd, you can't get from one to the other. This is a simple version of the '15 puzzle' - you can look that up for details.

how do I look it up? Sorry new at this

1. What is Puzzle 8 Configuration?

Puzzle 8 Configuration is a math logic problem that involves arranging the numbers 1-8 in a specific order using only three moves.

2. How do I solve Puzzle 8 Configuration?

To solve Puzzle 8 Configuration, you must use the three allowed moves (swapping two adjacent numbers, moving a number to an empty space, or moving a number over an adjacent number) to arrange the numbers 1-8 in the following order: 1, 2, 3, 4, 5, 6, 7, 8.

3. Can Puzzle 8 Configuration have multiple solutions?

Yes, Puzzle 8 Configuration can have multiple solutions. In fact, there are over 250,000 possible solutions for this puzzle.

4. Is there a specific strategy for solving Puzzle 8 Configuration?

Yes, there are several strategies that can be used to solve Puzzle 8 Configuration. One popular strategy is the "zero-sum" strategy, which involves creating a sum of zero in each row and column of the puzzle.

5. Is Puzzle 8 Configuration a real-world problem or just a math puzzle?

Puzzle 8 Configuration is primarily a math puzzle, but it also has real-world applications in fields such as computer science and game design. The puzzle helps to develop critical thinking and problem-solving skills.

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