We now search these sets for symmetrical board positions. This will give us all symmetric board positions which can be reached starting with the center vacant. Alternatively, we can take the complement of each board position—these are symmetric board positions where it is possible to finish with one peg in the center. We'll adopt the latter view in what follows.
There are seven possible types of symmetry for a configuration of pegs. The highest possible symmetry is square symmetry, in this case the board position is unchanged upon rotation by 90 degrees or reflection about the x or y axes (or either diagonal axis). There are 8 symmetry transformations which leave the board unchanged, forming the dihedral group D4. There are three order 4 subgroups of D4: 90 degree rotational symmetry, rectangular symmetry and symmetry about both diagonal axes, We also have three order 2 subgroups: 180 degree rotational symmetry, and reflection symmetry across one axis (orthogonal or diagonal axis). [In reality there are actually five order 2 subgroups, but the reflections form pairs. For example we have 34,500 board positions symmetric about the y-axis and another set of 34,500 which are 90 degree rotations of these, symmetric about the x-axis.]
If a board position has square symmetry, and it can be reduced to one peg, then this peg can only be at the center d4 or the equivalent holes from the rule of three: d1,a4,g4,d7. Why is this the case? It is because of the position classes. Suppose it is possible to finish somewhere other than the center. Then, because the board position is unchanged by 90 degree rotation, it must also be possible to finish at the same hole rotated through 90 degrees. But this will contradict the rule of three.
Similar reasoning shows that the same is true for the lower forms of symmetry, up until the last two. If we have only one reflection symmetry, we can finish on the axis of symmetry and not violate the rule of three.
|n (level)|||Sn|||Pegs||Complement Pegs||Type 1||Type 2||Type 3||Type 4||Type 5||Type 6||Type 7|
A count of boards with various symmetries. Type 1: square symmetry; Type 2: 90 degree rotational symmetry;
Type 3: symmetry across both diagonals; Type 4: rectangular symmetry; Type 5: 180 degree rotational symmetry;
Type 6: symmetry across x or y-axis; Type 7: symmetry across one diagonal axis
See a similar table for the the English 33-hole board
Of course, if only a few jumps are possible from a certain board position, the winning jump is usually the obvious one. The most difficult board positions would seem to be those having as many jumps as possible, but only one winning jump. For each n (or number of pegs p=32-n), we can identify such board positions. Sometimes there is even a unique board with p pegs with exactly one winning jump and as many total jumps as possible.
|Positions with as many jumps as possible but only one winning jump. † means the board position is unique.|
Above we see four positions found by this technique, with 10, 12, 17 and 22 pegs. All are solvable to one peg in the center, but only one starting jump leads to a solution.
These puzzles tend to be challenging to solve by hand. However, knowledge that there is a unique winning jump can help solve the puzzle. For example, after this first jump is executed, all jumps which could have been performed as a first jump must still be dead ends. The second jump can only be some jump opened up by the first jump, and so on. If you can identify the winning first jump, sometimes the rest of the solution follows more easily. Note that since these board positions have no symmetry, we could also consider finishing locations other than the center. Each finishing location produces a whole new set of puzzles, although many of them may be the same or similar to others.
Created in 2016. Last modified May 20th, 2016
Copyright © 2014 by George I. Bell