Four people were called up to the board to solve an 8-peice puzzle in a 3x3 grid. Two of the puzzles were impossible to solve.

#1 Number Puzzle (mixed)
 6  4  5
  
 1  7  8
 
 3  2  
 
A number puzzle is a puzzle in which you have to scramble these mixed up numbers into ones which are in order. In which the resulting puzzle should look something like this
 
#1 Number Puzzle (fixed, called "Home Position")
1  2  3
 
4  5  6
 
7  8
 
 
You make a move in this game by sliding numbers into the blank space, which moves around as the puzzle is played.  You want to get the puzzle from the starting position to the fixed, home position.  There is a simple algorithm you can follow to make this puzzle really easy:  When  looking at the puzzle as a 3 by 3 puzzle you can arrange the numbers that will go on the outside first
 
in puzzle #1 you can move the numbers 1, 2 and 3 into their correct spots and then numbers 4 and 7 into theirs by:

Step 1:  Try to get 1,2,3 into their correct spots by moving 1 into its spot then by putting 3 in the spot beside it. Once 3 is in the spot 2 should be in move 2 up to that spot, this will cause 3 to have to slide over into its correct spot. 
 
Step 2:  Now 1,2,3 are in their correct spots now try to get 4 and 7 into their spots by moving 7 into the number 4 spot under 1, so now once you get 4 beside 7, you can slide 7 down and put 4 in its right spot.
 
This is what it will look like
 
1 2  3
4 8  5
7 6
 
Step 3:  Now if you just rotate the numbers 8 ,5 and 6 you end up wiht a solved number puzzle.
 
Now, to scramble these puzzles, the teacher had picked up pieces of the puzzle, one in each hand, and swapped them before putting them back.  We discovered that switching a odd number of pairs will cause the puzzle to be unslovable but switching an even number of pairs will build a solvable puzzle. 


Net Diagram Theorem:  Given any net diagram, it is possible to add a line at any specified height to move any specified person into any specified note taker position.

Homework: Give proof for this theorem.