Peg solitaire (For the English board)


I did some kind of retrograde analysis on peg solitaire. I wrote a program which plays reverse solitaire. The program started with one peg and gained a peg on each turn. It searched for all the positions with two pegs. Then all the positions with 3 pegs. And so on, till it reached 32 pegs. All the positions it found were, of course, solvable.

Since I didn't want to save all the positions, I made a selection of puzzles. I saved those puzzles which are very symmetric.

It saved the puzzles with full orthogonal symmetry. (Vertical and horizontal symmetry) Like this one:
  ...  
  ...  
..ooo..
ooooooo
..ooo..
  ...  
  ...  

It also saved the puzzles with full diagonal symmetry. (Symmetric along both diagonals) Like this one:
  ...  
  oo.  
.oooo..
.oo.oo.
..oooo.
  .oo  
  ...  

And it saved the puzzles with full rotational symmetry. Like this one:
  ...  
  ..o  
.oooo..
..ooo..
..oooo.
  o..  
  ...  

The program found a total of 279 positions. I'm confident it found all of these positions, since the program searched through all positions. Or at least, it searched through all positions which can be solved with one peg in the middle of the board. I did a limited search through other positions as well. But it seems to be the case that all these very symmetric positions, if they are solvable, can always be solved with the last peg in the middle of the board. In fact, this can be explained. George Bell gave me an argument.

If a puzzle is solvable with the peg in the middle, the last peg in that puzzle can end in one of the following positions:
  .x.  
  ...  
.......
x..x..x
.......
  ...  
  .x.  

You can see that these crosses are always at a distance of 3 from each other. This has to be true and can be mathematically shown. It is explained on George Bell's page., which offers a lot of information about peg solitaire. There are also puzzles in which the last peg could end in one of the following positions:
  ...  
  x..  
.......
.......
..x..x.
  ...  
  ...  
Or
  ...  
  .x.  
.......
.......
x..x..x
  ...  
  ...  
Or reflections or rotations of these configurations. You may notice that these last two configurations are less symmetric. If we were able to reach a position in these configurations, we should also always be able to reach a position for the last peg on a position which is symmetric to that position in that configuration, if that puzzle has that same symmetry. So, for example, if we have a position with full orthogonal symmetry and we were able to land in that position for the last peg:
  ...  
  ...  
.......
.......
..x....
  ...  
  ...  
We should also be able to land in one of these positions:
  ...  
  ...  
..x....
.......
..x.x..
  ...  
  ...  
Because these positions are fully orthogonal to the previous position. And since the initial puzzle has full orthogonal symmetry, it could be solved in a symmetric way so the last peg would end in any of these positions. But you can notice that these positions are not three apart. The only configuration which has the necessary symmetry, is the one in which the last peg can end in the middle.

So if you ever wanted to play some different peg solitaire puzzles, you may challenge yourself with these 279 positions. There's the possibility to play these online too through George Bell's link.
Though, some of these positions are a bit repetitive since one position could be transformed to another very symmetric position.
If we delete all positions which can lead to another very symmetric position in 4 moves or less, only 129 positions remain.
If we delete all positions which can lead to another very symmetric position in 8 moves or less, only 90 positions would remain. This seems to be the most interesting collection to play to me.
If we delete all positions which can lead to another very symmetric position, only 76 positions remain.
You can right click on these links if you want to save the file for later use.

I wrestled my way through all positions. And a few of them are surprisingly hard. I saved these difficult positions here.
Here's an example of a difficult puzzle:
  .o.  
  .o.  
..ooo..
ooo.ooo
..ooo..
  .o.  
  .o. 


When in need for a solution, you may try a Peg Solitaire solver. I have used this one successfully.

I could have made an applet to play these puzzles online. But it's actually much more fun to play them on a real board. And one could create one's own board rather easily.



If you have puzzles of your own, i will gladly add them to this site.
Comments, puzzles and love mail may be sent to driesdeclercq@protonmail.com



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