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Peg Solitare

Peg Solitaire is a board game played on one with 33 cells. It looks like this:


Another version of the game uses a board with 37 cells rather than 33. And looks like the following:


The first number being the cellís position in a row of 7 cells and the second number being the row number labels the cells. The rows are counted from the bottom up. In the most basic puzzle, the board is set up with pegs in all cells except the centre one.

The object of this game is to make a series of jumps such that there is only 1 peg left out of the 32 and this peg must end in the centre (cell 44). A jump is when a peg is moved over an adjacent one and lands on an empty cell next to it. The peg that was jumped over is then removed from the board. The jumping sequence is similar to that of checkers except that instead of diagonal jumps, the jumps are made either horizontally or vertically.


A man named Mannis Charosh was able to come up with a way to analyse a given solitaire position to determine if it could be simplified into another given position. His method was done by applying a series of transformations to any starting position to see if it can be changed to the desired end position. If it is possible, then the positions are said to be equivalent. If they are not equivalent, then it is impossible to transform one position to another by the normal rules of solitaire. If they are equivalent, then it would be uncertain if the puzzle is solvable. This method gives any puzzle and on any board, a necessary but insufficient condition of possibility.

His method involves any three adjacent cells that are in a straight horizontal or vertical line. Where there are pegs in the cells, remove them and empty cells are to be filled. Thus if all three cells are filled, all three pegs are to be removed, and vice-versa.

If this method were to be applied to the classic problem of the one vacant cell in the middle, and we use cells 43, 44 and 45 (the three in the middle) as the set to test, the pegs in 43 and 45 will be removed while a peg would be placed in cell number 44. Therefore, a board with a vacancy in the centre is equivalent to a board with the cell 44 filled but cells 43 and 45 vacant. Therefore we would be able to conclude that the puzzle is not impossible, (which of course, we know). This method can be applied to all other puzzles.

Can we begin with a vacancy in 44 and end with a peg in 45? No, we cannot, since if we take cells 43, 44, and 45 again we cannot end up with a peg in 45 and the other two empty, neither would it hold true for 44, 45 and 46. (Refer to fig 2)

(Cells 43, 44 and 45)

(Cells 44, 45 and 46)

 

Fig 2

 

There are many other puzzles where solitaire is concerned, some of which are displayed below. They may use either type of board, discernible via the diagram.


The Latin Cross


Small Pyramid


Greek Cross


Fireplace, or Football Team


Big Pyramid


Shrine (Please note the ending piece is on cell 43, not the centre cell)


The Lamp


Normal


David Jump


Tilted Square


Corner to Corner


Octagram


Pentagram


Five Crosses


Double Cross

The diagrams from this point forth are representative of ending positions, not starting positions. You are to start with a vacant cell 44 and try to make the figure displayed in such cases.


Lonely Cross


Final Score


12 Guards

The solutions
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