# Peg solitaire solutions-How to Win the Peg Solitaire Game (English Board) (with Pictures)

Peg Solitaire Sailor's Solitaire is a very popular single player game played with a board having holes in the pattern of a plus sign. There are pegs in all the holes except one. The objective is to clear the board of all the pegs except one. To create this article, 9 people, some anonymous, worked to edit and improve it over time. Together, they cited 6 references.

Every move wolutions the parities of the three numbers equal to each other, so it is then impossible to finish with only a single peg. This really is a valid resource count because not only is, as we have just seen, any increase of A-X compensated for by a decrease in S, Peg solitaire solutions increase in S is only Spokane assessors tax by a jump from the middle which is compensated for by a decrease in A. A shorter and less technical version was published as a chapter in the book Mathematical Wizardry for Peg solitaire solutions GardnerAK Peters, [B5]. This is easily seen as follows, by an argument from Hans Zantema. In an efficient method for solving peg solitaire problems. The algorithm Peg solitaire solutions quite straight-forward:.

## Rope sheave. How to Solve Peg Solitaire (Hi-Q) with Step-by-Step Advice

Did this article Peg solitaire solutions you? This is an important move. Peg Solitaire Solution. Learn why people trust wikiHow. Diagonal jumps are not allowed, and there must always be an empty target hole on the other side of the peg being jumped in order for the move to be legal they can't jump over one peg and land on another. Not Helpful 17 Helpful To Sexual shave slave, let's take a quick refresher course on how a game of Peg Solitaire is played. This article has also been viewedtimes. Not Helpful 18 Helpful Now we want to go and rescue the lonely peg in the Peg solitaire solutions. In peg solitaire was completely solved on a computer using an exhaustive search through all possible variants.

Peg solitaire or Solo Noble is a board game for one player involving movement of pegs on a board with holes.

- Click here for hole Peg Solitaire Click here for hole Peg Solitaire Rules of the game: The hole version of peg solitaire consists of 33 holes see diagram below and 32 pegs.
- With easy-to-learn rules and seemingly simple goals, many are taken by surprise when they discover how deceivingly difficult the puzzle turns out to be.
- Detail of a French board marketed as Solitaire Di Venezia , solid ebony board with individually hand-blown glass marbles.

Detail of a French board marketed as Solitaire Di Venezia , solid ebony board with individually hand blown glass marbles. These pages were created and are maintained by George Bell gibell comcast. Bell snapshot of this site on recmath.

We can merely mention bean-bags, peg-boards, size and form boards, as some apparatus found useful for the purpose of amusing and instructing the weak-minded. The "Central Game".

A hand carved board with clay marbles, dating from the early 's photo courtesy St. John Stimson. This English board designed by Michael Graves has a unusual spiral marble trap. A French board marketed as Solitaire Di Venezia , solid ebony board with individually hand blown glass marbles. A hole board from India, ca. A French board made by C.

Jeandin, France, ca Photo courtesy the Slocum Collection. Hyper Solitaire, photo courtesy the Slocum collection.

Legal jumps in 4x4 toroidal solitaire. An 8-sweep? A 5-move solution to the a1-complement! Read the story behind the creation of this "very limited edition" board. John D. John H. Conway, Elwyn R.

Berlekamp, Richard K. George I. Bell, A Compendium of Peg Solitaire related papers, pages, 90 pages double sided.

This is a compilation of 13 papers relating to peg solitaire You can download all the papers separately, but this makes a nice, color-coded, 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. Bell and John D. DuPuy, D. Bell, Notes on solving and playing peg solitaire on a computer , last updated Robert A. Beeler and D. Joseph K. Barker and Richard E.

John Beasley , On hole solitaire positions with rotational symmetry , unpublished manuscript, Cut The Knot contains a good description of how to solve peg solitaire problems using block removals called packages and purges, Conway's terminology.

Jurgen Koller's Peg Solitaire web site even has ideas on how to construct your own board. MathWorld has a summary page with many printed references. Torsten Sillke has independently come up with Generalized Cross Boards. Sidney Cadot published an article in Dutch in the January issue of Machazine. In this article he describes how he was able to calculate all move solutions to the central game on the hole board.

He determined that there are exactly move solutions. Emmanuel Harang has a very extensive web page on the theory of the game. Unfortunately for me it is in French. Durango Bill has a page on peg solitaire where he also calculates the number 23,, This is the total number of board positions reachable from the central vacancy on the standard hole board not including positions equivalent by symmetry. He also calculates the total number of solutions to the central game. Gary Darby has created a Delphi program to solve peg solitaire problems, as well as an extensive site on other mathematical games.

The University of Waterloo has a games museum which contains some interesting history on peg solitaire. James Dalgety has one of the largest puzzle collections in the world. He runs puzzlemuseum. Erich Friedman has a nice collection of peg solitaire puzzles. John Robinson was an artist who in made a sculpture of the tree of knowledge in the form of a peg solitaire board, with pegs representing forbidden fruit. This site has some interesting ideas about the French board.

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 , where the game is described in detail. This French article is the earliest known printed reference to peg solitaire.

Top Accolades has a web page which descibes the shortest solution to the central game in a way that is easy to remember. 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. A wikihow page describing how to solve the central game on the English board.

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.

Durango Bill has a page on hole triangular peg solitaire. You win when there is only one peg left. Updated: March 29, Here is the complete game again: We clear the four side areas, come to the arrow and then to the final lying T-figure. It also received 16 testimonials from readers, earning it our reader-approved status.

### Peg solitaire solutions. How to Play Peg Solitaire

Generally you play this game without a strategy, so that you can't repeat the moves later. If you like to tell a solution, you have to establish a notation scheme.

If a peg is in the middle red , jumps in all directions are possible. Two jumps can be made from the blue field and one jump from the green one. There are 76 jumps altogether. You can make another classification of the holes of the field. There are four classes 4 colours. A peg always stays in the same class. This classification is very important in the theory of solitaire and play of solitaire. It is difficult to reach a given pattern. There is no solution to the European board with the initial hole centrally located, if only orthogonal moves are permitted.

This is easily seen as follows, by an argument from Hans Zantema. Divide the positions of the board into A, B and C positions as follows:. Initially with only the central position free, the number of covered A positions is 12, the number of covered B positions is 12, and also the number of covered C positions is After every move the number of covered A positions increases or decreases by one, and the same for the number of covered B positions and the number of covered C positions.

Hence after an even number of moves all these three numbers are even, and after an odd number of moves all these three numbers are odd. Hence a final position with only one peg cannot be reached, since that would require that one of these numbers is one the position of the peg, one is odd , while the other two numbers are zero, hence even. There are, however, several other configurations where a single initial hole can be reduced to a single peg.

A tactic that can be used is to divide the board into packages of three and to purge remove them entirely using one extra peg, the catalyst, that jumps out and then jumps back again.

Other alternate games include starting with two empty holes and finishing with two pegs in those holes. Also starting with one hole here and ending with one peg there. On an English board, the hole can be anywhere and the final peg can only end up where multiples of three permit. Thus a hole at a can only leave a single peg at a , p , O or C. A thorough analysis of the game is known.

A solution for finding a pagoda function, which demonstrates the infeasibility of a given problem, is formulated as a linear programming problem and solvable in polynomial time. A paper in dealt with the generalized Hi-Q problems which are equivalent to the peg solitaire problems and showed their NP-completeness. A paper formulated a peg solitaire problem as a combinatorial optimization problem and discussed the properties of the feasible region called 'a solitaire cone'.

In peg solitaire was completely solved on a computer using an exhaustive search through all possible variants. It was achieved making use of the symmetries, efficient storage of board constellations and hashing. In an efficient method for solving peg solitaire problems. One consequence of this analysis is to put a lower bound on the size of possible "inverted position" problems, in which the cells initially occupied are left empty and vice versa.

Any solution to such a problem must contain a minimum of 11 moves, irrespective of the exact details of the problem. It can be proved using abstract algebra that there are only 5 fixed board positions where the game can successfully end with one peg. The shortest solution to the standard English game involves 18 moves, counting multiple jumps as single moves:. The order of some of the moves can be exchanged. This solution was found in by Ernest Bergholt and proven to be the shortest possible by John Beasley in This solution can also be seen on a page that also introduces the Wolstenholme notation , which is designed to make memorizing the solution easier.

The only place it is possible to end up with a solitary peg is the centre, or the middle of one of the edges; on the last jump, there will always be an option of choosing whether to end in the centre or the edge. Following is a table over the number P ossible B oard P ositions of possible board positions after n jumps, and the possibility of the same pawn moved to make a further jump N o F urther J umps.

Since there can only be 31 jumps, modern computers can easily examine all game positions in a reasonable time. Note that the total number of reachable board positions sum of the sequence is 23,,, while the total number of possible board positions is 8.

### Peg solitaire - Wikipedia

The puzzle of peg solitaire is one consisting of a number of holes in a grid, some of which are filled with pegs. There are two classic setups, the English and European variants, as shown in Figure 1. Figure 1. The English left and European right setups of peg solitaire. One attempts to remove all pegs by moving pegs via jumps.

Jumps are always in either a vertical direction or a horizontal direction where one peg jumps over another into a hole, thereby removing the jumped peg. Two such moves on the English board are shown in Figure 2. Figure 2. Two moves, one horizontal jump and the other a vertical jump, in the English variation of peg solitaire. The objective in the English variant is to finish with a single peg in the center hole. In the European variant, it is not possible to finish with a peg in that location, but it is possible to finish with a peg in one of three other locations.

Figure 3. In a good attempt, this author can finish with two pegs left, although it is possible to end at a state where no further jumps are possible much earlier. Two sub-optimal final states are shown in Figure 4. Figure 4. Two sub-optimal final states of the English variant of peg solitaire. Such a puzzle is amenable to backtracking to find a solution. The algorithm is quite straight-forward:.

This algorithm is implemented in the source directory for both the English and European boards. The solution that is found is:. The default backtracking algorithm, unfortunately, has an exceptionally long run-time, due to the many possible choices one may make at each step. We may, however, exploit symmetry in the problem.

For example, the eight boards in Figure 5 represent essentially the same position. These boards are simply rotations or a flip followed by a rotation of each other. We will say that these boards are congruent to each other. A consequence of backtracking is that there is an absolute certainty that congruent boards will, never-the-less, be solved multiple times, even it is determined previously that the board does not lead to a solution.

The run-time of the default algorithm is significantly longer than this author was willing to wait hours. In order to exploit symmetry, there must be a means representing congruent boards by a unique identifier. One solution is to convert board to a bit number by traversing through the entries in each of eight orders corresponding to the eight symmetries, as shown in Figure 6.

We choose the number that is the smallest to represent this board, and all congruent boards will also result in the same unique identifier. In this case, that identifier is the fourth: All of the above boards shown in Figure 5 will map to this unique identifier.

Thus, we create a set of identifiers, and each time we determine that a board leads to a non-solution, its identifier is stored in a set, and if that board or any similar board is visited, the backtracking algorithm halts. Thus, this is a combine solution using backtracking and dynamic programming and This reduces the run time to 20 s. Peg solitaire Here, we will discuss a description of the peg solitaire puzzle , the backtracking algorithm , and an optimization.

The puzzle The puzzle of peg solitaire is one consisting of a number of holes in a grid, some of which are filled with pegs. The algorithm Such a puzzle is amenable to backtracking to find a solution. The solution that is found is: Jump from 3,5 to 3,3 , from 3,2 to 3,4 , from 3,0 to 3,2 , from 5,3 to 3,3 , from 3,3 to 3,1 , from 5,2 to 3,2 , from 4,0 to 4,2 , from 2,1 to 4,1 , from 2,3 to 2,1 , from 2,0 to 2,2 , from 2,5 to 2,3 , from 4,4 to 2,4 , from 2,3 to 2,5 , from 0,4 to 2,4 , from 0,2 to 0,4 , from 4,6 to 4,4 , from 2,6 to 4,6 , from 3,2 to 5,2 , from 1,2 to 3,2 , from 6,2 to 4,2 , from 3,2 to 5,2 , from 6,4 to 6,2 , from 6,2 to 4,2 , from 4,1 to 4,3 , from 4,3 to 4,5 , from 4,6 to 4,4 , from 5,4 to 3,4 , from 3,4 to 1,4 , from 0,4 to 2,4 , from 2,5 to 2,3 , and finally from 1,3 to 3,3.

An optimization The default backtracking algorithm, unfortunately, has an exceptionally long run-time, due to the many possible choices one may make at each step. Figure 5. Eight equivalent boards. Figure 6. Thus, these eight boards code the eight values: We choose the number that is the smallest to represent this board, and all congruent boards will also result in the same unique identifier.

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