Detail of a French board marketed as Solitaire Di Venezia, solid ebony board with individually handblown glass marbles. 

These pages were created and are maintained by George Bell (gibell@comcast.net) Last Modified June 4th, 2020 Copyright © 2006–2020 by George I. Bell 2016 snapshot of this site on recmath.org 


We can merely mention beanbags, pegboards, size and form boards, as some apparatus found useful for the purpose of amusing and instructing the weakminded. Albutt's "System of Medicine", 1899, VIII, 246 [B3] 
A 1697 engraving by CA Berey of the Princess Soubise beside a French solitaire board. 
This oneperson game (or puzzle) first appeared in France in the late 17th century. We know this because the puzzle is depicted in several art works of the period, most notably the engraving by Claude Auguste Berey shown on the right. The first mention of the game in print was in the French literary magazine Mercure Galant in 1697. Remarkably, this article escaped the notice of game historians for hundreds of years, and it came to light only in 2014, shortly after Mercure Galant became available on the internet. You can read more about this discovery in John Beasley's historical update [P10].

It seems everyone (in my generation) has run into this puzzle at some point. Boards range from drilled planks using golf tees, to beautifully crafted hardwood boards with indentations for marbles, including a nice rim around the edge for storage of marbles as they are removed. Computer versions of the game are also common (see my Javascript games below).
After unsuccessful attempts at a "manual solution", many people (myself included) try to write a simple computer program to solve it. Even if there are only 48 jumps available at each board position, this can lead to a prohibitively large number of possible jump sequences after only 15 jumps. This game is much older than the computer, and it is remarkable how much was proved about this game without the aid of computers.
Instead of trying to memorize a computer solution, or a YouTube video, the best way to remember a solution is to understand "block removals" or "packages", sequences of jumps that remove a whole block of pegs but leave the rest of the board untouched. Once you understand and have mastered the "3removal", "6removal" and "Lmove", the central game rapidly goes from frustrating to being quite simple (for details, jump ahead to here, or see [B1], [B3], [W2] or [W20]).
Lithographed Tin board with plastic pegs in two colors, © 2015 puzzlemuseum.com 
After the central game above has been solved, what then? We can try to find the "shortest" solution. All solutions contain exactly 31 jumps, because we start with 32 pegs and one is lost with each jump. However, when the same peg jumps one or more pegs, we call this one move. The question is, what is the solution with the least number of moves? This question may be rephrased more informally as: what is the solution that involves touching the smallest number of pegs? This question is more difficult than finding a solution to the central game, and we will return to it when discussing the English board.
The boards on this page have holes based on a square lattice; each hole has (at most) 4 neighbors. It is also possible to play on a triangular lattice, where each hole has (at most) 6 neighbors. I now have a separate page for Triangular Peg Solitaire. If you want some tips and solutions for the 15hole triangular board, check out my page on tips and solutions for the Cracker Barrel Puzzle. Below we will also discuss gridless boards.
James Dalgety [W13] is a puzzle collector who owns many peg solitaire boards. On his web page he discusses design faults of peg solitaire boards, which I include here for future puzzle designers. Common design faults include:
We mention quickly other possible notations: first, one can simply number the holes consecutively. This notation is quite common, the main problem with it is that the numbering changes every time the board shape changes. John Conway invented an alphabetic notation which mirrors the symmetry of the board [B3]. The most general notation is 2D Cartesian coordinates, with the origin at the center of the board (to preserve symmetry), or at the lower left corner (to avoid negative coordinates). This notation (see below, right) is bulky and is primarily used in computer programs.

For larger boards, "Standard 7x7 notation" must be extended. "Standard 9x9 Notation" is the obvious extension, where the columns are ai and the rows 19 (the most interesting boards have an odd width, and that is why we go from 7x7 to 9x9). An unfortunate aspect of this notational switch is that the central hole goes from "d4" to "e5".
In standard 7x7 notation, to refer to a jump we simply list the starting and ending holes separated by a dash, i.e. "e5e3" for one of the jumps available on the left, and "e5e3c3a3" is shorthand for the move consisting of the jumps "e5e3, e3c3, c3a3".
A hand carved board with clay marbles, dating from the early 1900's (photo courtesy St. John Stimson). 
A board is called rectangularsymmetric if it is unchanged when reflected about the x or yaxes, and rotationallysymmetric if it is unchanged by any 90 degree rotation. A board is called squaresymmetric if it is both rectangularsymmetric and rotationallysymmetric. Finally, a squaresymmetric board with a unique central hole is called odd because its width is odd, and it also has an odd total number of holes. We see that the board above is odd, squaresymmetric and gapless.
The general peg solitaire problem is to play from a full board with one peg missing to a board position where only one peg remains. These problems are called single vacancy to single survivor problems, sometimes abbreviated SVSS. The special problem where the initial vacancy and survivor are the same board location is called the (single vacancy) complement problem.
Consider a diagonal labeling of the board holes (in two ways) as shown below:

This English board designed by
Michael Graves has a novel spiral marble trap. 
At the standard starting position with only the center hole vacant, you can easily check that N_{1}=N_{3}=N_{4}=N_{6} =11, N_{2}=N_{5}=10, and the total number of pegs T=32. Therefore the 6 starting parities (TN_{i}), i=1,2, ... 6 and therefore the position class of the board is (Odd, Even, Odd, Odd, Even, Odd). All board positions reachable from this starting position must be in this same position class. The position class of a board position with only a single peg is easy to calculate, it is odd on all diagonals except for the two the peg is in. Hence we see that the only possible finishing locations for the game must be the intersections between diagonals 2 and 5, or the board locations (0,0)=d4, (3,0)=g4, (0,3)=d1, (3,0)=a4 or (0,3)=d7.
Consider the board position with every hole filled by a peg. Then T=33, N_{i}=11 for all i, and all six parities are Even. The empty board with no pegs also lies in the same position class: (Even,Even,Even,Even,Even,Even). This is the defining property of a nullclass board: any board position and its complement are always in the same position class (the complement of a board position is the board position where each peg is replaced by a hole and vice versa). It is important to realize that nullclass boards are special, and not all boards are nullclass boards.
A French board marketed as
Solitaire Di Venezia,
solid ebony board with individually hand blown glass marbles. 
From a practical standpoint, how do we determine whether a board
is nullclass or not?
The most obvious technique is to label the board in the
above fashion and count the number of 1's, 2's, 3's through 6's.
If these six numbers have the same parity (all odd or all even) then the board
is nullclass, otherwise it is not.
A more clever technique is to apply local transformations
to the full board that do not change the position class we are in,
and try to reduce it to the empty board.
A simple class of very useful transformations is to
take the complement of any three consecutive board locations
(vertically or horizontally).
A solitaire jump itself is such a transformation, but there are others, such
as removing three pegs in a row,
or replacing two pegs separated by a hole by a peg in the hole.
Using this technique we can discover which position
class any pattern of pegs is in, and which finishing holes are possible.
Figuring out possible finishing holes from the starting position. 
Figuring out possible finishing holes from a failed game. 
The transformations in the last section are useful to find all possible finishing holes from any configuration of pegs on any board (nullclass or not). Consider, for example, the peg solitaire puzzle to the right, from the iPhone App Puzzlium [W23]. In order to determine where the final peg can be, I mentally erase lines of three pegs (denoted by the red bars). This leaves only one peg, so the only possible finishing holes are the two marked "F". The finishing peg in the "F" hole must never be captured, because no other peg can finish there.
We can use this technique not just at the start, but at any time during a solution. One can attempt to solve the puzzle, and failing that, observe carefully the final pattern of pegs. Using the same transformations, reduce it to a single peg, which will give the possible finishing holes. The simplest case is two pegs a distance 2 apart, in which case the finish must be the hole between them (or any hole a multiple of 3 away in any direction). Here is a summary of local transformations for use in determining finish holes:
When determining the finishing hole, one can also ignore the board during the process, and not worry when your transformation results in a peg off the board. Of course, in the end a finishing hole can only be one that is on the board. Can you determine the possible finishing holes from the failed game shown on the right?
How can the knowledge of finishing holes help solve a puzzle? You can keep track of the pegs which can finish, being careful not to remove all of them. In addition, on cramped boards you may be able to figure out the final jumps, and aim for such a board position. On the first Puzzlium board shown, this is particularly useful.
What happens if the local transformations fail to reach a one peg state? This can only happen if all pegs can be removed, leaving an empty board. This shows the original puzzle is not solvable to one peg! Puzzlium [W23] has internal checks which prevent users from entering unsolvable peg solitaire puzzles.
On the left we see a board position on Wiegleb's board
from which a remarkable 16loop can be executed.
This loop move fits on the English Board,
however on Wiegleb's board this board position can actually be reached in a
single vacancy to single survivor game.
An interesting puzzle which can be solved by hand is to
find a solution with a 16loop.
Backward play is hard to comprehend, because our brain does not easily interchange the concepts of "hole" and "peg". It is hard to shake the perception that a hole is the absence of a peg. It is much easier to understand "backward play" by realizing that it is the same as forward play from the complement of the current board position.
For example, suppose we want to solve the puzzle mentioned above on
Wiegleb's board.
Namely, we want to find a solution beginning with one peg missing, and ending
with a spectacular 16loop.
Playing forward, how can we possibly know which 27 jumps might end at the complex
17peg pattern shown above?
It seems hopeless, even impossible;
but now we know that playing the game backwards from the 16loop position
is the same as playing the game forward from the complement of this position.
So we begin from the complement of the 16loop position
(shown on the left).
If we can solve this board to one peg
(and this is possible),
then the 16sweep is ours!
We then begin with a board filled except for where this solution ended,
and play the jump sequence in reverse.
This will magically reproduce the 16loop board position!
We will return to this puzzle later.
In [B3] this technique is called the
"time reversal trick",
and it can be a powerful tool for solving certain problems.
A 33hole board from India, ca. 1830 © 2015 puzzlemuseum.com 
Over the years, literally hundreds of versions of this board have been produced under brand names such as Puzzle Pegs, HiQ, and Classical Solitaire. Even today, if you google peg solitaire board you will find dozens of English boards to choose from. The most expensive boards, with exotic marbles, usually include the four extra holes of the French Board.
Screen capture of "Pegged". 
Why is the English board so popular? It is the smallest squaresymmetric, gapless board on which the central game is solvable (see [P3]). In fact, it is the smallest such board on which every complement problem is solvable.
Ernest Bergholt found an 18move solution to the central game in 1912. In 1964, John Beasley proved that there is no shorter solution to the central game [B1]. Since then the problem of finding minimal length solutions has been attacked by many people using a computer, and minimal length solutions to all single vacancy to single survivor problems have been found. In 2012, Joseph Barker and Richard Korf [P8] applied advanced search techniques to this problem—their search algorithm can find Bergholt's solution using less than 3 seconds of CPU time! They can find shortest solutions to all 21 SVSS problems using less than 2 minutes of CPU time [P8].
Over the years, there have been various attempts to describe a solution to the central game that is easy to remember. Bergholt's 18move solution is shortest, but tends to be difficult to recall. Nonetheless, a specialized "Wolstenholme notation" has been invented to assist in recalling it [W19]. Another solution, called "Jabberwockey", is given in Martin Gardner's book [B2]. This solution has some nice symmetry properties, which make it easier to remember, as well as faster because at a certain point one can capture pegs with both hands simultaneously. Click here to see a diagram of this solution (paired jumps are shown in red).

This leaves you in the heartshaped board position shown on the upper right. Notice the six jump loop starting from d5: d5b5b3d3f3f5d5. This leaves the board with a "T" shaped configuration of pegs on the right which can be solved (by inspection): d4f4, d6d4, c4e4, f4d4. The animation on the right shows the whole solution, or see this diagram, or this WikiHow Page [W21]. Note that the screen capture of "Pegged" (above left), is actually one frame of a 2007 YouTube video by Kevin Paul Scarrott. The video shows the same solution, but reflected about the xaxis.
If diagonal jumps are allowed, what is the shortest solution to the central game? In Beasley's book [B1] he gives a 16move solution, and remarks that it is not known if this is the shortest. In 2006, I was able to complete the exhaustive search, and the shortest solution has 15 moves [P4].
A web page on the shortest solution when
diagonal jumps are allowed.
A web page with sample computer calculations for the English 33hole board.
Symmetric English Puzzles,
a collection of 279 symmetric positions which can be solved to one peg in the center.
A French board made by C. Jeandin, France, ca 1890. Photo courtesy the Slocum Collection. 
This board is not a nullclass board, therefore no complement problem is solvable on it. Using the position class theory, one can prove that if you begin from a filled board with only the center vacant, it is impossible to finish with one peg. To show this, simply compute the six parities of the starting board position. All six parities are even, and therefore all six parities will always be even. But any one peg position has four odd parities and two even parities, so no single peg position can be reached.
Solitaire was very popular in France at the start of the 18th century. Below we show a shield or escutcheon bearing a French peg solitaire board (see [P10]). This is part of a French coat of arms registered in 1709. Peg solitaire is possibly the only puzzle to have appeared in a traditional coat of arms.
There are ten single vacancy to single survivor problems solvable on this board, and each can be solved in 20 or 21 moves. If you are solving a problem by hand, the best technique to use (on any board) is to decompose it into block removals. Note that the French board decomposes nicely into L and 3removals as shown on the right. In order to solve a particular problem, one can modify this diagram with appropriate starting and ending moves, as shown on the left.
In each diagram, the location of the starting hole is marked by an "S", and the finishing hole by an "F". After dividing the board up into block removals, one must also check that the catalyst needed by each block removal is present. The sequencing of the block removals is indicated by the numbering. Click on each diagram to show the full solution played out. If you don't understand how to turn such a diagram into a solution, read the introduction to block removals at [W2].
If the game begins with the center peg missing, this board position is in the position class of the empty board, so it is not possible to finish with one peg. An interesting modification of the rules which allows for a d4 to d4 solution is known as Cremers' Key [W16]. According to [W16], this concept was invented around 1998 by Frans Cremers, a retired teacher from Aalter, Belgium. Starting with the center vacant, play proceeds as usual, except that the player is allowed to replace the central peg once (when the hole is unoccupied). This puts the board position in the correct position class, and a solution can be obtained. The shortest solution to the central game using Cremers' Key has 20 moves.
An escutcheon, ca. 1709 "Jaque Chavillot" [P10]. 
Note that Cremers' key works not only from the center start, but from any start. In the general case one is allowed to replace the peg at the starting hole once. The goal is to finish with one peg, not at the starting location, but always in the center. The reason why this works is clear if you understand position classes. On the French board, the board with every hole filled by a peg is in the same position class as the board with one peg in the center. Therefore, when you replace a peg at the starting hole, what you are doing is changing the position class to that of the full board, so that finishing in the center is now possible. The results of [P3] show that Cremers' Key will put the board in the position class of the center finish on any gapless, odd, squaresymmetric board that is not nullclass.
Another way to make the central game solvable is to allow diagonal jumps. In this case the central game can be solved in as little as 13 moves. One interesting puzzle with diagonal jumps is to begin with pegs at all locations except for the central 9 holes {c3, c4, c5, d3, d4, d5, e3, e4, e5}, and try to play to the complement of this position with only the center 9 holes occupied. Allowing diagonal jumps, this "big central game" is solvable, but the same problem is not solvable on the 33hole English board. The shortest possible solution to the "big central game" has 13 moves. One reason the "big central game" is interesting is that using it one can solve any SVSS problem [P4].
A web page on the shortest solution when diagonal jumps are allowed
Symmetric French Puzzles,
a collection of 428 symmetric positions which can be solved to one peg in the center.
The 6x6 board supports some remarkably short solutions. John W. Harris and Harry O. Davis studied this board in the 1960's, and found that most single vacancy to single survivor problems can be solved in 15 moves, with a few cases requiring 16 moves [W1].
This board does have one unusual property: it is possible for any peg on the board to reach some corner. No matter where the final peg is to be left, it is possible to arrange things so that the last four moves start from the four corners. If this board is easy for you try finding solutions with this property. Click here to see an elegant 15move solution with this property, due to John Harris [W1]. Here is another one I found. It is not always possible to have a minimal length solution with this corner finish property.
A double vacancy complement problem
is a puzzle where two pegs are removed at the start,
and your goal is to finish with two pegs in the original vacancies.
On the 6x6 board, all 93 double vacancy complement problems are solvable
(unlike the English 33hole board,
where four double vacancy complement problems are not solvable
[B1, p. 1069]).
It is tedious to find solutions for the 93 cases,
but all are solvable (here is a
sample solution).
The 6x6 board is not the smallest board where all double vacancy complement problems are solvable,
the 6x4 rectangular board also has this property.
This board is a member of a general class known as a draughtsboard. Such boards are obtained by taking any square board, and labeling the holes alternately as on a chess or checkers board. Then the board is rotated 45 degrees and the black squares define the holes of the draughtsboard. The 41hole diamond board can be so obtained starting from a 9x9 square board.
You can try to find a center to center solution using the Cremers' Key [W16] rule modification, but you will not succeed. Replacing the center peg does give one a board position in the correct position class, but using the resource count shown below one can prove that it is not possible to finish with one peg.
Several interesting variations to the 41Hole Diamond Board have been proposed. Around 1882, H.A.H. Hermary proposed removing the leftmost and rightmost holes from the board, giving Hermary's 39Hole board, shown on the right. This board is nullclass, rectangularsymmetric, and most (but not all) complement problems are solvable [B1]. The central game on Hermary's 39hole board can be solved in a minimum of 23 moves.
In 1894, A. Huber removed 4 holes to produce Huber's 37Hole board, shown on the left.
This board is nullclass and
squaresymmetric—the central vacancy and other
complement problems are solvable.
The central game on Huber's 37hole board can be solved in
a minimum of 20 moves.
We note that the complement problem at the "tip of the arm" is not solvable
(proved in [P3]).
This board has a similar size as the standard English board, but it has 14 corners rather than 8. In 1941, B. M. Stewart showed that all 25 SVSS problems were solvable on this board. My program has found that all of them are solvable in 1719 moves, with only the d1 (top hole) complement requiring 19 moves. The d4complement can be solved in a minimum of 18 moves.
A classic problem on a checkers board is to begin with 24 pegs
along the edges with the 4x4 center section empty.
It is not hard to show that this board position is in the position class of the empty board,
hence it cannot be solved to one peg [W26].
Cool Moves, 2007 
In "Hoppers", Nob Yoshigahara came up with a whole set of challenge cards for this board. He also had the idea of including a special red frog which must be preserved and perform the final jump. The original name for Nob's version was "Marsh Madness".
If you rotate this board 45 degrees, it
can be seen as a 13 hole Diamond Board, or
draughtsboard created from a 5x5 square board,
with the addition of diagonal jumps along both diagonals.
Without diagonal jumps, no
single vacancy to single survivor problem
on this board is solvable.
However with the diagonal jumps many problems become solvable,
and its small size makes solutions easier to work out.
An interesting advanced problem is to try to solve the
central game on this board in as few as
7 moves.
A web page with strategy tips for this board.
A Soli2 board. 
There are 36 different SVSS problems on Wiegleb's Board, all are solvable except for the (4,0) or e1 complement problem. A proof that the (4,0) or e1complement is unsolvable is not easy. An outline of a proof is given in Beasley's book [B1], but filling in the details is nontrivial (I asked John Beasley about this, and he agrees that filling in the details is not easy). I have found an alternate proof by integer programming techniques [P3]. A computer proof by exhaustive search is possible, but time consuming.
Trying to solve the (4,0) or e1complement problem is several orders of magnitude harder than any problem on the English board. If you have a fast peg solitaire solver try testing it on this problem. If you want to try a problem that actually has a solution, try any other complement problem on this board, or try the (4,0) or d1complement problem on the 3232 Board.
Problems on this board can be solved most easily using block removals as shown on the figures to the left. The numbering shows the ordering of the block removals. Numbers subdivided a, b, c are block removals that must be interleaved, in other words part b begins before part a is finished. Click on each diagram to see the full solution played out. Beasley [B1] gives a block removal diagram for one other complement problem on this board.
Handmade Wiegleb's Board in painted plywood, 16 mm marbles, DIY Puzzles. 
No other combination of starting and finishing holes can contain a 16sweep, except of course rotations or reflections of the above three problems. The three problems above can be solved in a minimum of 22, 24, and 23 total moves, respectively. For solutions, see [P2] or this web page.
The solution catalog shows the shortest length solution to all 35 solvable problems on this board, an effort in 2004 which required 3 months of CPU time on a 1 GHz Pentium PC. The results can be found in our paper [P2], and in 2012 were confirmed by Barker and Korf [P8]. All the problems are solvable in a minimum of 2023 moves, with only the (3,0) or e2 complement requiring 23 moves. The central game can be solved in 22 moves, but no fewer. There are a few problems on this board with unique minimal length solutions, up to symmetry and move order. For example, there is a unique 20 move solution from c4 to i4.
A web page of computational results for Wiegleb's Board
A web page on 16loops on Wiegleb's Board
(print it out and try them yourself)
Using the same technique [W1] as for the 6x6 board, it is easy to prove that the solution to any SVSS problem must have at least 24 moves. In 1986, John Harris found a 25move solution by hand [W1]; finding a shorter solution is a difficult computational task, but in 2014 I found a 24move solution (and here is another).
The longest finishing sweep on this board has length 21.
It is not difficult to find a solution finishing with a
21sweep—this is a fun exercise to work out by hand.
As usual, you must begin from the complement of the sweep position
(one possibility is shown on the right).
On this board, what is the shortest solution to the central game? This is currently an open question. In 2004, Alain Maye found a solution by hand in 34 moves. It is likely that the shortest solution has around 30 moves, by analogy with the 10x8 board, which is of similar size. Curiously, the odd side length makes the techniques used on the 10x8 board much less powerful.
An interesting challenge is to demonstrate that the 32loops shown to the left can be reached from single vacancy starts. Some of them can also be the final move to a complement problem. It is also possible to find a solution to the central game ending with a 30loop.
with links to shortest solutions. 
Rectangular boards where both n and m are even
are more interesting than you might think,
we call them "eveneven boards".
Why are eveneven boards interesting?
By analogy with the proof [W1] on the 6x6 board,
it is easy to prove that the solution to any
SVSS problem
on an eveneven board must have at least
The grid to the left summarizes my results on eveneven boards. The green squares denote nullclass boards, otherwise the square is red. Square boards appear along the diagonal. The tiny 4x4 board has a minimum solution length of 8 moves, an 8 move solution does not exist, although 9 moves is possible. For many larger eveneven boards, the minimum solution length seems to be attainable.
On the 6x4 board it is not difficult to find 11move solutions, and we have already mentioned that the 6x6 board has 15move solutions. On the 8x6 board, I have found 19move solutions, and on the 8x8 square board, 24move solutions. All of these are automatically the shortest possible, by the above argument.
In 2014, I found a 29move solution on the 10x8 board.
In 2015, I found a 34move solution on the 12x8 board!
This is the largest board (96 holes) for which a solution having minimal length is known.
Finding these shortest solutions on such large boards requires advanced search techniques,
I use A* search
based on the number of filled Merson regions [B1].
67hole Siege Board, ca. 1880 
I investigated these boards in 2003 and found there are exactly 12 generalized cross boards where the complement problem is solvable at every board location. Of these 12, the only board with square symmetry is the standard 33hole board. Two have rectangular symmetry (top row of figure to the left), three have mirror symmetry about one axis, two have diagonal symmetry (second row) and the remaining four have no symmetries at all.

A web page on Generalized Cross Boards
Three challenges on the 3232 Board
(print it and try them yourself)
A Solo Board. Photo courtesy Jaap Scherphuis [W20]. 
The Solo board is not nullclass. For detailed analysis of this board see this web page on Jaap's Puzzle Page [W20].
There is now a separate web page for Triangular
Solitaire.
Jumping rules leading to a universal board indicate a fundamental change in the game—the position classes have "collapsed" into a single set, really the whole concept of a position class becomes meaningless. Since every board position is in the same position class, all boards are nullclass (this term has also become meaningless). A universal board might sound like a good thing, but it usually means that the puzzle is too complicated.
Consider, for example, the English Board. From a randomly chosen board position (probability of 1/2 to have a peg in any hole) there are an average of 9.5 jumps available. If diagonal jumps are allowed, the board becomes universal, with an average of 17.0 jumps available. From experience, it seems too difficult to consider 17 possible jumps from each board position.
Star Jump, by Creative Crafthouse

Hyper Solitaire, photo courtesy the Slocum collection 
Clock Solitaire, 2017 
Solomon, by Kadon Enterprises

Clock Solitaire, photo courtesy St. John Stimson 
Peg solitaire need not be played on a regular grid. In principle it can be played on any network of intersecting lines (or even an arbitrary graph [P7] if one allows jumps along any triple of connected nodes). We call boards which are not based on a regular (square or triangular) grid "gridless". To be considered "gridless", either the holes do not lie on lattice points, or the jumps are not parallel to lattice lines. Boards like Hoppers which allow diagonal jumps have been mentioned above, but they really should be classified here. Our derivation of nullclass does not extend to gridless boards; however, the concept can sometimes be extended using "invariant parity counts", as explained in Beasley's book [B1, p. 635 and p. 2412]. Several examples of invariant parity counts are given below.
The most common gridless board seems to be Star Jump (shown to the left), a 10hole board sold by Creative Crafthouse, it has also been sold under the names Penta, Star Solitaire, and Star Trekker. One marble is removed at the start and this puzzle uses normal peg solitaire rules, with jumps allowed along the lines on the board. As usual, the goal is to finish with one peg.
Let N_{1} be the number of pegs in the inner ring. A peg solitaire jump can never change the parity of N_{1}. If we begin from a vacancy in the inner ring, then N_{1}=4 (even). The only one peg positions with N_{1} even have N_{1}=0 with one peg in the outer ring. Thus, if we start in the inner ring, we can only finish in the outer ring (and viceversa). It turns out all such problems are solvable. Star jump can be extended to a 16hole board with 5fold symmetry. One such board is given in Beasley's book [B1, p. 2423], another option is shown in this blog [W24].
Solomon was invented by Martin Gardner around 1970 as a two player game, and can also be used for peg solitaire. This 19hole board in the shape of a hexagram can be purchased from Kadon Enterprises.
Some gridless boards have no corners, a property which we have not seen to this point. Hyper Solitaire is a 33hole English board which has been warped so that jumps can be made "around the corners". Although the number of possible jumps has increased from 76 to 108, Hyper Solitaire is not universal, as shown by the following argument: consider the 12 hole set R marked in red on the photo to the left. We note that there is no jump which can add or remove one peg from R (pegs can only be removed from R in pairs). Therefore, the parity of the number of pegs in the set R can never change. Thus if the initial vacancy is in R, you can only finish in R, and if you start outside R, you can only finish outside R.
Round Solitaire is a modification of Hyper Solitaire invented in 2009 by Tetsuro Kawahara, he removed the outer ring of 12 holes. The central game is solvable on this board in a minimum of 8 moves. Round Solitaire is not universal [P10], this can be demonstrated using the set R of the previous paragraph.
The brass board with clay marbles on the right was probably made in the early 1900's, the engraving seems to read "H.S.H. (PROT)ECTED". Another copy can be found in item #3663 of the Jerry Slocum collection [W22]. It is not known if this board was meant for playing peg solitaire, and this seems unlikely to me because the marbles are so close together. If viewed as a peg solitaire board, the jumping rules are not obvious. It can be interpreted as Hexagon(3) with additional jumps around the outer ring, giving 66 possible jumps. With this 66 jump set, the board becomes universal. John Beasley has investigated possible jumping rules and has found a certain minimal set of 42 jumps which make a nice peg solitaire board. He calls it "Clock Solitaire" [W18]. For an analysis of this board see [P11].
This is not an exhaustive list of gridless boards,
more can be found in Beasley's book [B1],
or his more recent update article [P10].
Legal jumps in 4x4 toroidal solitaire 


An 8loop? 

A 5move solution to the a1complement! 
A toroidal chess board! 
Consider, for example, the 4x4 board. Under normal peg solitaire rules, this board isn't very interesting, the only SVSS problems which are solvable have the form: vacate a2, finish at a3 (or d3). Under toroidal jumping rules, the board becomes universal. The symmetry of the board is also very different: there are no corners or edges—all holes are the same. The red graphic to the left shows a curious 8loop on the 4x4 toroidal board. Unfortunately, it cannot be reached from any single vacancy start. My program has found that all SVSS problems on this board can be solved in a minimum of 5 moves (see the example solution to the left). It is remarkable that the first move can be a double jump!
The reader may enjoy demonstrating that the central game on 3x3 toroidal solitaire is solvable. In fact, it is difficult to lose this game, is there any dead end? An analysis of n by m rectangular boards under toroidal jumping rules reveals that if n and m are both multiples of 3 the board is still nullclass and the "rule of three" still applies. If both n and m are not multiples of 3, the board is probably universal. A simple extension of the rule of three gives this, because by wrapping around the board any pair of holes can be considered to have coordinates a multiple of 3 apart (when n and m are not multiples of 3). If one of n and m is a multiple of 3 and the other is not, then we have an intermediate case where the rule of three applies in only one dimension (the one divisible by 3). My program has found that the shortest solution to the complement problem on the 5x5 toroidal board has 8 moves, and on the 6x6 toroidal board, 11 moves.
Starting from any rectangular board, we can also identify opposite edges in other orientations to play peg solitaire on a cylinder, Möbius strip, or Klein bottle. It is not clear if these geometries have any more to offer than the torus.
A proof that the 4x4 toroidal board is universal.
"Pegged", an early computer version of peg solitaire. 
"Never Lose" Triangle(5) (2014). 
Most computer versions of peg solitaire have the ability to take back moves, all the way back to the beginning if necessary. In my opinion, this tends to make a puzzle seem easier, compared with solving on a mechanical board. Resetting the board is also trivial for a computer puzzle. Computer versions of peg solitaire can also include demos or solutions, and they can point out bad or good jumps.
Here are links to online peg solitaire puzzles I have created. You must have JavaScript activated in your browser to play them.
I have also designed the (hexagonal) levels of a free online peg solitaire game. You will need to download Shockwave to use it. Thanks to Rob Gordon of Article19 for the GUI.
Given a board and a (solvable) single vacancy to single survivor problem, there is a minimum number of moves that can solve it. These solutions have an elegant look to them and they tend to be extremely hard to find by hand.
The table below shows a list of boards, together with some statistics about each. If you click on a board, you will see another table listing all single vacancy to single survivor problems solvable on that board, together with information about these solutions. These boards are squaresymmetric (except for Diamond32) and we only list unique single vacancy to single survivor problems. In other words, if one problem can be obtained from another by rotation and/or reflection, only one will be listed. If you keep clicking on the tables, you can view diagrams of minimal length solutions.
Some column heads you will see that require explanation:
Board Name [click to see catalog] 
Number of Holes 
Null Class? 
Number of Problems 
Longest Sweep (any problem) 
Minimal Length Solution Properties  

Solution Lengths  Longest Sweep  Time to Calculate  
Diamond32  32  Yes  35  8 or 9  1719  8  2 hours 
English  33  Yes  21  9 †  1519  8  2 hours 
6x6  36  Yes  21  10 ‡  1516  10  6 hours 
French  37  No  10  9 †  2021  9  24 hours 
Diamond41  41  No  4  9  26  9  3 hours 
Wiegleb's  45  Yes  35  16 ‡  2023  14  3 months 
Table Footnotes: (†) From John Beasley's book [B1]. (‡) This is the longest sweep geometrically possible on this board, and at least one such sweep can be realized as the final move to a single vacancy to single survivor problem. 
The information in the above table has been calculated by other authors in the case of the English, French and 6x6 boards. My results have been checked against their results. In 2012, Barker and Korf [P8] confirmed my results for the 41hole diamond and 45hole Wiegleb's board.
Read the story
behind the creation of this "very limited edition" board. 
A good introductory reference which has a nice progression of problems is the 2007 book by Koetke [B4]. This book starts out with small boards that are easy to solve, and discusses the problems encountered for larger boards. It also has example programs in java. This is one of the few peg solitaire references that contains a lot of detail about solving the game computationally.
If you try to solve a peg solitaire problem in an inefficient manner your program can take forever to run, even on the standard 33hole board. For example the most obvious technique is to store the sequence of moves (or jumps) and try to exhaustively go through all possible sequences (generally using a depthfirst search). Because there are a large number of move sequences that result in the same board position, such an algorithm is extremely inefficient. Somewhat surprisingly, this inefficient algorithm may still quickly find a solution to the central game, but we seek algorithms which perform well even in the worst case. One significant improvement is to use a hash table or some other means to store board positions seen previously so you do not have to investigate them farther.
A similar technique is to only keep track of the set of boards at each level in the tree, rather than the moves. I call this a search by levels. This is much faster than a straight search over move sequences and can solve any problem on the 33hole board relatively quickly. For boards larger than this, additional techniques are needed.
I have used four ideas to speed up a search by levels:
Finally, one should be careful not to confuse the computer's failure to find a solution with a proof that no solution exists. These calculations are complex and lengthy, particularly when resource counts, symmetry and forward/backward calculations are all being used. Logical bugs in the code can easily prevent the computer from finding a solution, and much testing is required to make sure the results are reasonable. For example, no program that reports "no 17 move solution to the English board central game" should be trusted unless it can find the Bergholt solution in 18 moves. You can also test your program by reproducing these tables.
For details on computational search techniques see [P4] and [P8].
The most useful resource counts generally have negative values in corners (in fact, it is not hard to show that a resource count can only have a negative value at a corner). If a certain move leaves the board with a value that is less than that of our final position, we know that this move cannot possibly lead to a solution. A very useful resource count on the 41hole diamond board is:

The above resource constraint can be applied on the French board (by ignoring any value that is not on that board) and it is still somewhat useful. It can also be used on the English board, but with only a very minor speed increase.
On Wiegleb's board a moderately useful resource count (for computations) has a "1" at the eight corners and "+1" at the 12 interior locations: d2, f2, b4, d4, f4, h4, b6, d6, f6, h6, d8 and f8 (and zeros everywhere else), see a diagram here.
"Culture", a vertically oriented board with hanging pegs © 2013 John Robinson 
Consider a rather arbitrary board without any interior holes (gapless). This board might be quite large, and in the simplest case could be square. We will try to quantify the difficulty of three different problems:

While this 13peg puzzle is difficult to solve by hand, a computer can easily solve it using exhaustive earch. However, imagine a similar problem with, say, 50 pegs.
Like problem #1, problem #2 also asks if you can reduce a pattern of pegs to a single peg. Does this mean problem #2 is NPComplete also? No, it does not, because a complement problem does not start from an arbitrary pattern of pegs. The starting position has every hole occupied, except for the starting vacancy, hence it is a very regular pattern. Clearly if the board is not nullclass, there is no solution to problem #2 or #3. It is very easy to check if a board is nullclass, so we don't need to require this. As the difficult case we may as well assume all boards from now on are nullclass.
Suppose we consider problem #2 on an arbitrary rectangular nullclass board (with both edges at least 4). Finding a solution to a complement problem on such a board, I claim, is actually very easy, and can probably be solved in linear time. Why? Because this problem is easy to solve by hand using block removals. You simply visually identify blocks of pegs you need to remove and make sure the required catalyst is present and that after the block removal you don't strand any pegs too far away from the core bunch. You also need to be careful near the edge of the board. This logic could be programmed, and would result in a computer algorithm that could solve complement problems on rectangular boards extremely quickly.
Even boards which are "not too different" from a rectangular board should also be easy. It is rather difficult to quantify "not too different", but basically any gapless nullclass board that doesn't have any tight spaces can be solved using block removals. The standard 33hole board should fit into this category. It easier to complete this argument on triangular boards, and I now have a simple algorithm that can solve any single vacancy problem on a triangular board of arbitrary size.
In summary, problem #1 has been proven NPComplete. I believe problem #2 is not very hard on "well behaved" boards, which include at least rectangular boards (and certainly triangular boards). Problem #3 is very difficult on any board with more than about 50 holes, and I believe no algorithm can find shortest solutions in polynomial time. In 2012, Joseph Barker and Richard Korf wrote a paper on the subject of searching for short solutions [P8].
B1.  John D. Beasley, The Ins & Outs of Peg Solitaire, Oxford Univ. Press, 1985 (the paperback Edition 1992, contains an additional page: Recent Developments) ISBN 019286145X (paperback).  
B2. 
Martin Gardner,
The Unexpected Hanging and Other Mathematical Diversions,
University Of Chicago Press, 1991 (paperback), pp. 122135
ISBN 0226282562.
[The chapter "Peg Solitaire" is based on a
Scientific American column that appeared in the June, 1962 edition.]
Recently updated with republished as Martin Gardner, Knots and Borromean Rings, RepTiles, and Eight Queens, Cambridge University Press, 2014, ISBN 0521758710.  
B3.  John H. Conway, Elwyn R. Berlekamp, Richard K. Guy, Winning Ways for Your Mathematical Plays, Volume 4, AK Peters, 2004 (second edition), Chapter 23, "Purging Pegs Properly".  
B4.  Walter Koetke, Classic Thinking Games with Java, Basics & Beyond, Inc., 2007.  
B5.  George I. Bell, Peg Solitaire with Diagonal Jumps, in Ed Pegg Jr., Alan H. Schoen, Tom Rodgers (Editors) Mathematical Wizardry for a Gardner, AK Peters, 2009.  
B6.  George I. Bell, A Compendium of Peg Solitaire related papers, 2014 (175 pages, 90 pages double sided). This is a compilation of 13 papers relating to peg solitaire (20032014). You can download all the papers separately, but this makes a nice, colorcoded, bound volume. To see the list of papers, download the Table of Contents. Please email me if you would like a zip file containing all 13 papers (2.6 MB). 
P1.  Ryuhei Uehara and Shigeki Iwata, Generalized HiQ is NPcomplete, Trans. IEICE, E73, pp.270273, 1990  
P2.  George I. Bell and John D. Beasley, New Problems on Old Solitaire Boards, Board Game Studies #8, presented for the 8th international colloquium on board game studies, Oxford, England in April 2005 (finally published in Board Game Studies Online in 2014).  
P3.  George I. Bell, A Fresh Look at Peg Solitaire, Mathematics Magazine, 80:1628, 2007.  
P4.  George I. Bell, Diagonal Peg Solitaire, INTEGERS Electronic Journal of Combinatorial Number Theory, Vol 7, G1, 2007.  
P5.  G. DuPuy, D. Taylor, Using Board Puzzles to Teach Operations Research, INFORMS Transactions on Education, Volume 7, Number 2, January 2007.  
P6.  George I. Bell, Notes on solving and playing peg solitaire on a computer, last updated 2014.  
P7.  Robert A. Beeler and D. Paul Hoilman, Peg Solitaire on Graphs, Discrete Math. 311: 21982202, 2011.  
P8.  Joseph K. Barker and Richard E. Korf, Solving Peg Solitaire using Bidirectional BFIDA*, in the Proceedings of the TwentySixth AAAI Conference on Artificial Intelligence, Toronto, 2012.  
P9.  John Beasley, On 33hole solitaire positions with rotational symmetry, unpublished manuscript, 2012.  
P10.  John Beasley, An update to the history of peg solitaire, 2014 (paper based on a presentation at the 34th International Puzzle Party in London, UK on August 8th, 2014).  
P11.  George I. Bell, Clock Solitaire, Cubism For Fun #101, November 2016.  
P12.  George I. Bell, Designing peg solitaire puzzles, Recreational Mathematics Magazine, No. 7, pp. 519, 2017; DOI 10.1515/rmm20170011 
W1.  Solitaire (pdf version), Issue #28 (web version) of The Games and Puzzles Journal, September 2003.  
W2.  Cut The Knot contains a good description of how to solve peg solitaire problems using block removals (called packages and purges, Conway's terminology).  
W3.  Jurgen Koller's Peg Solitaire web site even has ideas on how to construct your own board.  
W4.  MathWorld has a summary page with many printed references.  
W5.  JC Meyrignac's web site contains results from computer runs on the French Board, as well as a peg solitaire competition and other items.  
W6.  Torsten Sillke has independently come up with Generalized Cross Boards.  
W7.  Sidney Cadot published an article (in Dutch) in the January 1998 issue of Machazine. In this article he describes how he was able to calculate all 18move solutions to the central game on the 33hole board. He determined that there are exactly 936 18move solutions.  
W8.  Emmanuel Harang has a very extensive web page on the theory of the game. Unfortunately (for me) it is in French.  
W9.  Durango Bill has a page on peg solitaire where he also calculates the number 23,475,688. This is the total number of board positions reachable from the central vacancy on the standard 33hole board (not including positions equivalent by symmetry). He also calculates the total number of solutions to the central game.  
W10.  Gary Darby has created a Delphi program to solve peg solitaire problems, as well as an extensive site on other mathematical games.  
W11.  The University of Waterloo has a games museum which contains some interesting history on peg solitaire.  
W12.  Michel Schellekens is an Associate Professor at the National University of Ireland with an interest in peg solitaire. (http://www.cs.ucc.ie/~mpcs/ link broken as of 2015)  
W13.  James Dalgety has one of the largest puzzle collections in the world. He runs puzzlemuseum.com, there you will find history and photos of his extensive solitaire collection.  
W14.  Erich Friedman has a nice collection of peg solitaire puzzles.  
W15.  John Robinson (19352007) was an artist who in 1993 made a sculpture of the tree of knowledge in the form of a peg solitaire board, with pegs representing forbidden fruit.  
W16.  This site has some interesting ideas about the French board.  
W17.  Rob Stegmann has a vast puzzle web site with a nice page on Peg Solitaire and Jumping Puzzles.  
W18.  The John and Sue Beasley Website has some interesting recent observations of the game. This includes a transcription from the French magazine Mercure Galant from 1697, where the game is described in detail. This French article is the earliest known printed reference to peg solitaire.  
W19.  Top Accolades has a web page which descibes the shortest solution to the central game in a way that is easy to remember.  
W20.  Jaap Scherphuis has a peg solitaire web page, with a nice discussion on block removals and how to determine which problems are solvable on various boards. Jaap also has a Peg Solitaire Java Applet.  
W21.  A wikihow page describing how to solve the central game on the English board.  
W22.  The Jerry Slocum Mechanical Puzzle Collection is one of the largest puzzle collections in the world, and includes many rare and unusual peg solitaire boards.  
W23.  The iPhone App Puzzlium has thousands of peg solitaire puzzles, you can solve them or submit new puzzles.  
W24.  Imre Polik, How to solve puzzles? Peg solitaire with optimization (2015), discusses solving a new pentagonal board using SAS optimization software.  
W25.  Taishi Oikawa, Kazuaki Yamazaki, Tomoko Taniguchi, and Ryuhei Uehara have created a Peg Solitaire Font. They presented a paper on the new font at the Bridges 2017 conference in Canada.  
W26.  David Percy, Editorial, February 2015 talks about various peg solitaire puzzles on a checkers board. 
Copyright © 2006–2020 by George I. Bell