# Math Logic problem

## 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...

## The Attempt at a Solution

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Mark44
Mentor
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.

Dick