Peg Solitaire
Contents of this Page
What is Peg Solitaire?
Solutions
Mobility of the Pegs
A Pattern at the Beginning
A Pattern in the End
Program
Building of a Solitaire Game
Other Boards
To the History
References.
Peg Solitaire on the Internet
.
To the Main Page     "Mathematische Basteleien"

What is Peg Solitaire?
Peg solitaire is an old board game for one player. 
A game board with 33 holes in cross form is given. 32 pegs are in. The centre hole is empty. You have to remove the pegs one after the other by jumping horizontally or vertically over one. In the end one peg should be left in the centre hole. 

The game is called solitaire (Latin solus = alone) because of the only player.

The game solitaire is called jumper, too. The name is not unique. A solitaire can also be a card game for one player or a piece of jewelery having a single diamond.


Solutions    top
Removing all pegs but one is not easy to solve.
... 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. 
E.g. you can number the holes in lines. "15-17" means you jump peg 15 to 17 and remove peg 16. 
1st solution: 
15-17, 28-16, 21-23, 07-21, 16-28, 31-23, 24-22, 21-23, 26-24, 23-25, 32-24-26, 33-25, 26-24, 12-26, 27-25, 13-27, 24-26, 27-25, 10-12, 25-11, 12-10, 03-11, 10-12, 08-10, 01-09-11, 02-10, 17-05, 12-10, 05-17.

You need 31 jumps, because you must remove 31 pegs. Double jumps like (32-24-26) or (01-09-11) counts as one move. So the solution above has 29 moves.

The board is symmetrical horizontally, vertically or diagonally, therefore there are seven more solutions.
2nd solution:
The "world record" is a solution with 18 moves by E. Bergholt from 1912.
15-17, 28-16, 21-23, 24-22, 26-24, 33-25, 18-30, 31-33-25, 09-23, 01-09, 06-18-30-28-16-04, 07-21-23-25, 13-11, 10-12, 27-13-11, 03-01-09, 08-10-12-26-24-10, 05-17.
3rd solution:
The following solution is clever. You use the L-move four times and get the house figure (the house stands on its head). Then you take down this figure with a six-jump move.
...... 05-17, 08-10, 01-09, 03-01, 16-04, 01-09, 28-16, 21-23, 07-21, 24-22, 21-23, 26-24, 33-25, 31-33, 18-30, 33-25, 06-18, 13-11, 27-13, 10-12, 13-11, 24-26- 12-10-08-22-24, 17-15, 29-17, 18-16, 15-17. 


Mobility of the Pegs  top
..... 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. 
The moves on the board are more complicated because you don't know which moves will be used or which ones will be repeated in the same place. 


...... 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. (See more in book 2).

A Pattern at the Beginning top
Getting rid of as many pegs in patterns as possible is a simple way of playing solitaire. One peg must be left over somewhere. These simple puzzles are good for beginners to learn to play the big solitaire.
Solutions:
1) Submarine: 10-24, 15-17, 24-10, 19-17, 10-24.
2) Greek Cross: 10-02, 24-10, 19-17-05, 02-10, 15-17, 10-24, 29-17.
3) Pyramid 16: 18-06-04, 26-12, 09-11, 12-10, 24-26, 27-25, 23-09, 04-16, 10-24, 25-23, 16-28, 21-23, 28-16, 15-17.


The following game goes from the big diamond via the small diamond to the centre position. 
Solution: 11-03, 09-07, 23-31, 25-27, 07-21, 03-01, 27-13, 31-33-25, 13-11, 01-09, 21-23, 15-17, 18-16-04-06-18, 19-17, 24-22, 10-24, 25-23, 22-24, 29-17. 

A Pattern in the End top
You start with 33 pegs and finish with a pattern.
...... It is difficult to reach a given pattern. For an example I choose a square of four pegs.  ......
Solution: 29-17, 26-24, 17-29, 32-24, 23-25, 31-23, 22-24, 25-23, 33-25, 08-22-24, 10-08, 07-09, 21-07, 02-10-08, 07-09, 12-10, 03-11, 24-26, 27-25, 19-17, 10-24, 13-27, 24-26, 27-25, 09-23, 01-09.

You have more fun in looking for new (symmetrical)  patterns. 
Solutions: 
2 squares: 19-17, 16-18, 14-16, 05-17-15, 04-16-14, 29-17, 18-16, 06-18, 07-09, 13-11, 18-06, 16-04, 01-09, 03-11, 30-18, 28-16, 21-23, 27-25, 16-28, 18-30, 33-25, 31-23.

Avenue: 19-17, 30-18, 17-19, 20-18, 27-25, 15-17, 04-16, 17-15, 07-09, 14-16, 06-04, 18-06, 03-11, 01-03, 28-30, 16-28, 25-23, 33-25, 31-33, 23-31, 21-23, 09-01, 11-09, 13-11.

Pair: 19-17, 30-18, 17-19, 06-18, 13-11, 27-13, 26-12, 18-06, 03-11, 01-03, 04-06, 16-04, 11-09, 13-11, 08-10, 11-09, 03-11, 22-08-10, 11-09, 04-16, 24-22, 21-23, 07-21, 32-24-22, 31-23, 16-28, 21-23-31.


Program      top
There are many transformations of the board game solitaire to the computer on the internet. Anyway I wrote another program for studying. I missed a game, which took the moves down. 
The program is simple. If you find the string "oo." and mark the first sign "o" with the mouse pointer, then "Oo." is set. If you find the string "Oo." and mark the sign "." with the mouse pointer, then "..o" is set. The moves are written down at the same time. This is enough for playing. If you set a wrong "O" by mistake, you can replace the "O" with an "o" clicking on the small "o" on the right. The incorrect small large "O" will become small again. So you may make the correct selection. - Sorry, a back function is missing. 
I offer the game as Download. It is written in Visual Basic and needs VBRUN300.dll.


Program Steckhalma
...... I recommend to explore the solitaire game by the program "Steckhalma" made by Otto Drews. 
The special attribute is that you can reverse the steps. Please go to "Ein Zug zurück" (= One step backwards). 

The program is written in Visual Basic 2010 Express. You may download it  by courtesy of the author.

It consists of two zipped files.
Steckhalma.exe und SteckhalmaLösung.ebt.


Building of a Solitaire Game       top
There is hardly a household in my surrounding, which has no solitaire game. Nevertheless I will shortly explain, how to build a game.
(Drawing 1) In the simplest case you draw the game pattern on paper and use any tokens. 

(Drawing 2) This is the standard version of the game. You draw the cross pattern on a thick piece of wood and drill holes with the same depth. You succeed in making the same holes with a "depth feeler" on the drill. Then you must produce  pegs with the same length (or you take dowels), which must lightly stick in the holes. 

(Drawing 3) You can drill through the board. Then you must make pegs using two dowels of different diameters. 

(Drawing 4) You can drill hemispheres (or half rounds) and lay marbles in it. 


Other Boards  top
The game with 33 holes described up to now is the standard game at least in Germany. It is called the English solitaire. 

French Solitaire
...... There is also a game with 37 holes, which offers more possibilities of playing. It is called French solitaire. 
There is no solution for starting with an empty hole in the centre and finishing with the last peg being in the center hole. - There is a solution starting with a hole above on the left and ending with one peg below on the right. (2)
Solution: 03-01, 12-02, 08-06, 02-12, 19-06, 32-19, 30-32, 36-26, 05-07, 18-05, 20-18, 33-20, 01-11, 18-05, 04-06, 07-05, 20-07, 09-11, 16-18, 15-13, 22-20, 29-27, 18-31, 05,18, 35-25, 18-31, 20-33, 07,20, 37-27, 20-33, 34-32, 23-25-35, 26-36, 35-37.



Dreieckssolitär
...... The triangle solitaire has the solution
06-01, 04-06, 01-04, 07-02, 13-04, 02-07, 11-04, 14-05, 10-03, 03-08, 04-13, 12-14, 15-13.


Solitaire at the Palm
......
......
Solution: 25-23, 9-24, 17-15, 35-17, 18-16, 36-18, 15-17, 18-16, 23-25, 33-35, 16-34, 35-33, 13-15, 4-13, 6-4, 1-7, 3-1, 13-4, 1-7, 7-9, 9-24, 11-13, 29-11, 10-12, 28-10, 13-11, 10-12, 31-29, 40-31, 42-40, 43-37, 45-43, 31-40, 43-37, 12-30, 29-31, 38-23, 23-21, 37-22, 21-23, 23-25, 39-24, 25-23 

(George Bell's solution)

Today (July 20111) you may ask: What is the Palm?


Square 5x5 Solitaire
...... ...... Lösung:
08-18, 06-08, 03-13, 01-03, 04-02, 14-04, 05-03, 03-01, 16-06, 01-11, 18-08, 11-13, 08-18, 22-12, 23-13, 12-14, 15-13, 25-15, 10-20, 20-18, 13-23, 24-22, 21-23

To the History    top
The origin is shrouded in mystery as in many old puzzles. Solitaire is said to have been invented by a French aristocrat while in solitary confinement in the Bastille in the 18th century. Since the 19th century the game solitaire fascinated people in new variations. 


References   top
(1) Pieter van Delft, Jack Boltermanns: Denkspiele der Welt, München 1980
(2) John D. Beasley: The Ins and Outs of Peg Solitaire, Oxford 1985
(3) Theo Hartogh: Brettspiele aus aller Welt neu entdeckt, Niedernhausen/Ts. 2000


Torsten Sillke sent the following list of references for further studies.

- W Ahrens;
  Mathematische Unterhaltungen und Spiele, Band I.
  Leipzig, 2. Aufl., 1910, pp 182-210

- E R Berlekamp, J H Conway, R Guy;
  Winning Ways,
  for your mathematical play, Vol 2, (Part 4: Diamond)
  Academic Press, London, 1982
  chapter 23 Puring Pegs Properly, pp 697-734
  (German: Gewinnen, Strategien für mathematische Spiele,
   Band 4: Solitairspiele, Vieweg Verlag, Braunschweig, 1985)

- John D. Beasley,
  "The Ins & Outs of Peg Solitaire",
  Oxford Univ. Press, 1985
  (paperback Edition 1992, contain an additional page: Recent Developments)
  ISBN 0-19-286145-X  (paperback)

- Bialostocki, Arie
  An application of elementary group theory to central solitaire.
  College Mathematics Journal. (May 1998) v. 29(3) p. 208-212.
  ZblD 1999b.01301
  uses the Klein 4-group.

- N. G. de Bruijn;
  A Solitaire Game ans Its Relation to a Finite Field,
  Journal of Recreational Mathematics (1972) v. 5(2) p. 133-137

- Pieter van Delft, Jack Botermans;
  Denkspiele der Welt,
  Hugendubel (1977)
  (orig: Creative Puzzles of the World)

- Jörn Eichler, Jochen Jäger, Thomas Ludwig;
  Spielverderber - Solitaire mit dem Computer l\"osen,
  c't 1999 Heft 7, 218-225

- Martin Gardner,
  The Unexpected Hanging and Other Mathematical Diversions
  Simon & Schuster (1968)
  Chapter 11: Peg Solitaire. p122-135 
  the 33-hole solitaire and the 6-square.
  (Logik unter dem Galgen, Vieweg, 1971, Kapitel 11: Patience mit Figuren)

- Kurt-Heinz Grube;
  Zum Solitairespiel,
  MU (Der Mathematikunterricht) 26:2 (1980) 37-63
  ZblD 1980x.01194
  - Reiss' 'three-pile-game' , explains the 16 fundermental classes
  - solves several special figures and the standard problem.

- Irvin Roy Hentzel;
  Triangular Puzzle Peg,
  Journal of Recreational Mathematics (1973) v. 6(4) p. 280-283

- Irvin Roy Hentzel, Robert Roy Hentzel;
  Triangular Puzzle Peg,
  Journal of Recreational Mathematics (1986) v. 18 p. 253-256
  - solving the order 5 triangle

- E. Lucas;
  Recreations Mathematiques, Vol I,
  2nd edition, 1891, pp 87-141

- Miller, W.A.;
  Triangular peg solitaire on a microcomputer.
  Journal of Recreational Mathematics (1991) v. 23(2) p. 109-115.
  ZblD 1992d.00202
  - BASIC program

- M. Reiss;
  Beiträge zur Theorie des Solitär-Spiels,
  Crelles Journal 54 (1857) 344-379

- Jerry Slocum, Jack Botermans;
  Puzzles Old and New, U. Wash. Press, 1987,
  (German: Geduldsspiele der Welt, Hugendubel, 1987)
  Abbildungen von alten Solitairspielen gibt es in 
  Weiter sind eine Reihe von Aufgaben und ein Dreieck mit Spitzen.

- Schwartz, Benjamin L. and Ahlburg, Hayo;
  Triangular peg solitaire,
  Journal of Recreational Mathematics. (1983) v. 16(2) p. 97-101
  - solving the order 5 triangle

- Schwartz, Benjamin L.
  A solitaire pebble game.
  Journal of Recreational Mathematics. (1994) v. 26(3) p. 161-167.
  ZblD 1995d.02383

- B. M. Stewart;
  Solitaire on a Checkerboard,
  American Mathematical Monthly, 48 (Apr 1941) 228-233
  (A paper which gives solutions to all the single-vacancy
  single-survivor problems on the 32-hole draughtsboard.)


Peg Solitaire on the Internet  top

German

Senia Heckner 
Solitario Game  (Dreieckssolitär online)

Wikipedia
Solitär (Brettspiel)



English

Alexander Bogomolny
Peg Solitaire  (a java applet to play the 33-hole solitaire)

Charly Founès 
Peg out 1, Peg out 2 (2 Applets)

George Bell
Peg Solitaire

Gary Darby 
Peg Solitaire Game

Japanese Web site
Firsts and Lasts of Peg Solitaire

Jean-Charles Meyrignac
Peg Solitaire (136 problems) (Applet)

Mark Williams
PegSol v 1.1

MazeWorks
Peg Solitaire  (Applet)

Wikipedia
Peg solitaire

youtube video
Lösungsweg Solitair


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