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 yaxis and another set of 34,500 which are 90 degree rotations of these, symmetric about the xaxis.]
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.








The above table shows one sample board position of each type, followed by the total count of such board positions. Note that each board position is included in only the highest symmetry type (we do not count square symmetric positions in the lower symmetry types). In addition, we have removed all symmetric board positions which can already be reached on the English Board. You can now play the 279 positions of Type 14 on this Symmetric French Positions Javascript game.
n (level)  S_{n}  Pegs  Complement Pegs  Type 1  Type 2  Type 3  Type 4  Type 5  Type 6  Type 7 

0  1  36  1  0  0  0  0  0  0  0 
1  1  35  2  0  0  0  0  0  0  0 
2  3  34  3  0  0  0  0  0  0  0 
3  15  33  4  0  0  0  0  0  0  2 
4  70  32  5  0  0  0  0  0  1  1 
5  341  31  6  0  0  0  0  0  2  3 
6  1604  30  7  0  0  0  0  0  8  8 
7  6950  29  8  0  0  0  0  2  22  17 
8  27948  28  9  0  0  3  0  3  59  31 
9  102261  27  10  0  0  0  1  14  133  93 
10  335839  26  11  0  0  0  1  29  283  148 
11  984710  25  12  0  1  3  3  49  591  309 
12  2558220  24  13  0  1  9  3  96  1092  512 
13  5858375  23  14  0  0  0  4  140  1973  879 
14  11789357  22  15  0  0  0  8  253  3204  1388 
15  20795984  21  16  1  1  9  12  296  4910  2058 
16  32106854  20  17  2  3  13  12  489  6815  2627 
17  43386122  19  18  0  0  0  17  485  8892  3215 
18  51362742  18  19  0  0  0  14  752  10617  4154 
19  53371113  17  20  2  4  14  16  588  11829  3965 
20  48801369  16  21  5  2  24  20  852  12434  4732 
21  39361771  15  22  0  0  0  21  576  11803  3684 
22  28039820  14  23  0  0  0  22  752  10687  4064 
23  17646892  13  24  1  4  16  17  398  8763  2228 
24  9813533  12  25  3  4  17  19  509  6872  2729 
25  4808524  11  26  0  0  0  13  214  4905  1175 
26  2068047  10  27  0  0  0  20  290  3269  1450 
27  776914  9  28  0  3  4  9  92  1982  424 
28  253243  8  29  1  3  11  9  101  1169  521 
29  70245  7  30  0  0  0  5  28  588  79 
30  16690  6  31  0  0  0  5  32  288  169 
31  3350  5  32  1  0  1  1  4  123  22 
32  536  4  33  1  1  2  3  6  42  32 
33  62  3  34  0  0  0  2  1  17  0 
34  11  2  35  0  0  0  1  0  3  3 
35  0  1  36  0  0  0  0  0  0  0 
Total  374349517  17  27  126  258  7051  113376  40722  
Table 1: 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 yaxis; Type 7: symmetry across one diagonal axis See a similar table for the the English 33hole 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=32n), 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.
You can now play all ?? puzzles with 4 to 27 pegs (finishing in the center) on my Difficult French Positions Javascript game.
Created in 2016. Last modified May 20th, 2016
Copyright © 2014 by George I. Bell