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Dots and Boxes is a pencil-and-paper game for two players (sometimes more). It was first published in the 19th century by French mathematician Édouard Lucas, who called it la pipopipette.^[1] It has gone by many other names,^[2] including the game of dots,^[3]dot to dot grid,^[4]boxes,^[5] and pigs in a pen.^[6]

The game starts with an empty grid of dots. Usually two players take turns adding a single horizontal or vertical line between two unjoined adjacent dots. A player who completes the fourth side of a 1×1 box earns one point and takes another turn. (A point is typically recorded by placing a mark that identifies the player in the box, such as an initial.) The game ends when no more lines can be placed. The winner is the player with the most points.^[2][7] The board may be of any size grid. When short on time, or to learn the game, a 2×2 board (3×3 dots) is suitable.^[8] A 5×5 board, on the other hand, is good for experts.^[9]

The diagram on the right shows a game being played on a 2×2 board (3×3 dots). The second player ('B') plays a rotated mirror image of the first player's moves, hoping to divide the board into two pieces and tie the game. But the first player ('A') makes a sacrifice at move 7 and B accepts the sacrifice, getting one box. However, B must now add another line, and so B connects the center dot to the center-right dot, causing the remaining unscored boxes to be joined together in a chain (shown at the end of move 8). With A's next move, A gets all three of them and ends the game, winning 3–1.

Strategy[edit]

For most novice players, the game begins with a phase of more-or-less randomly connecting dots, where the only strategy is to avoid adding the third side to any box. This continues until all the remaining (potential) boxes are joined together into chains – groups of one or more adjacent boxes in which any move gives all the boxes in the chain to the opponent. At this point, players typically take all available boxes, then open the smallest available chain to their opponent. For example, a novice player faced with a situation like position 1 in the diagram on the right, in which some boxes can be captured, may take all the boxes in the chain, resulting in position 2. But, with their last move, they have to open the next, larger chain, and the novice loses the game.^[2][10]

A more experienced player faced with position 1 will instead play the double-cross strategy, taking all but 2 of the boxes in the chain and leaving position 3. The opponent will take these two boxes and then be forced to open the next chain. By achieving position 3, player A wins. The same double-cross strategy applies no matter how many long chains there are: a player using this strategy will take all but two boxes in each chain and take all the boxes in the last chain. If the chains are long enough, then this player will win.

The next level of strategic complexity, between experts who would both use the double-cross strategy (if they were allowed to), is a battle for control: An expert player tries to force their opponent to open the first long chain, because the player who first opens a long chain usually loses.^[2][10] Against a player who does not understand the concept of a sacrifice, the expert simply has to make the correct number of sacrifices to encourage the opponent to hand him the first chain long enough to ensure a win. If the other player also sacrifices, the expert has to additionally manipulate the number of available sacrifices through earlier play.

In combinatorial game theory, dots and boxes is an impartial game and many positions can be analyzed using Sprague–Grundy theory. However, Dots and Boxes lacks the normal play convention of most impartial games (where the last player to move wins), which complicates the analysis considerably.^[2][10]

Unusual grids and variants[edit]

Dots and Boxes need not be played on a rectangular grid – it can be played on a triangular grid or a hexagonal grid.^[2]

Dots and Boxes has a dual graph form called 'Strings-and-Coins'. This game is played on a network of coins (vertices) joined by strings (edges). Players take turns cutting a string. When a cut leaves a coin with no strings, the player 'pockets' the coin and takes another turn. The winner is the player who pockets the most coins. Strings-and-Coins can be played on an arbitrary graph.^[2]

A variant Kropki played in Poland allows a player to claim a region of several squares as soon as its boundary is completed.^[11]

In analyses of Dots and Boxes, a game board that starts with outer lines already drawn is called a Swedish board while the standard version that starts fully blank is called an American board. An intermediate version with only the left and bottom sides starting with drawn lines is called an Icelandic board.^[12]

A game called Trxilt combines some elements of Dots and Boxes with some elements of Chess.

References[edit]

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1. ^Lucas, Édouard (1895), 'La Pipopipette: nouveau jeu de combinaisons', L'arithmétique amusante, Paris: Gauthier-Villars et fils, pp. 204–209.
2. ^ ^abcdefgBerlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (1982), 'Chapter 16: Dots-and-Boxes', Winning Ways for your Mathematical Plays, Volume 2: Games in Particular, Academic Press, pp. 507–550.
3. ^Holladay, J. C. (1966), 'A note on the game of dots', American Mathematical Monthly, 73 (7): 717–720, doi:10.2307/2313978, JSTOR2313978, MR0200068.
4. ^Swain, Heather (2012), Play These Games: 101 Delightful Diversions Using Everyday Items, Penguin, pp. 160–162, ISBN9781101585030.
5. ^Solomon, Eric (1993), 'Boxes: an enclosing game', Games with Pencil and Paper, Dover Publications, Inc., pp. 37–39, ISBN9780486278728. Reprint of 1973 publication by Thomas Nelson and Sons.
6. ^King, David C. (1999), Civil War Days: Discover the Past with Exciting Projects, Games, Activities, and Recipes, American Kids in History, 4, Wiley, pp. 29–30, ISBN9780471246121.
7. ^Berlekamp, Elwyn (2000), The Dots-and-Boxes Game: Sophisticated Child's Play, AK Peters, Ltd, ISBN1-56881-129-2.
8. ^Berlekamp, Conway & Guy (1982), 'the 4-box game', pp. 513–514.
9. ^Berlekamp (2000), p. xi: [the 5×5 board] 'is big enough to be quite challenging, and yet small enough to keep the game reasonably short'.
10. ^ ^abcWest, Julian (1996), 'Championship-level play of dots-and-boxes'(PDF), in Nowakowski, Richard (ed.), Games of No Chance, Berkeley: MSRI Publications, pp. 79–84.
11. ^Grzegorzka, Jakub; Dyda. 'Dots - rules of the game'. zagram.org. Retrieved 2017-11-27.
12. ^Wilson, David, Dots-and-Boxes Analysis Results, University of Wisconsin, retrieved 2016-04-07.

External links[edit]

• Barile, Margherita. 'Dots and Boxes'. MathWorld.
• Ilan Vardi, Dots Strategies.

The pigpen cipher (alternatively referred to as the masonic cipher, Freemason's cipher, Napoleon cipher, and tic-tac-toe cipher)^[2][3] is a geometric simple substitution cipher, which exchanges letters for symbols which are fragments of a grid. The example key shows one way the letters can be assigned to the grid.

Security[edit]

The use of symbols instead of letters is no impediment to cryptanalysis, and this system is identical to that of other simple monoalphabetic substitution schemes. Due to the simplicity of the cipher, it is often included in children's books on ciphers and secret writing.^[4]

History[edit]

The cipher is believed to be an ancient cipher^[5][6] and is said to have originated with the Hebrew rabbis.^[7][8] Thompson writes that, “there is evidence that suggests that the Knights Templar utilized a pig-pen cipher” during the Christian Crusades.^[9][10]

Parrangan & Parrangan write that it was used by an individual, who may have been a Mason, “in the 16th century to save his personal notes.”^[11]

In 1531 Cornelius Agrippa described an early form of the Rosicrucian cipher, which he attributes to an existing Jewish Kabbalistic tradition.^[12] This system, called 'The Kabbalah of the Nine Chambers' by later authors, used the Hebrew alphabet rather than the Latin alphabet, and was used for religious symbolism rather than for any apparent cryptological purpose.^[13]

Variations of this cipher were used by both the Rosicrucian brotherhood^[14] and the Freemasons, though the latter used the pigpen cipher so often that the system is frequently called the Freemason's cipher. Hysin claims it was invented by Freemasons.^[15] They began using it in the early 18th century to keep their records of history and rites private, and for correspondence between lodge leaders.^[3][16][17] Tombstones of Freemasons can also be found which use the system as part of the engravings. One of the earliest stones in Trinity Church Cemetery in New York City, which opened in 1697, contains a cipher of this type which deciphers to 'Remember death' (cf. 'memento mori').

George Washington's army had documentation about the system, with a much more randomized form of the alphabet. And during the American Civil War, the system was used by Union prisoners in Confederate prisons.^[14]

Example[edit]

Using the Pigpen cipher key shown in the example above, the message 'X MARKS THE SPOT' is rendered in ciphertext as

Variants[edit]

The core elements of this system are the grid and dots. Some systems use the X's, but even these can be rearranged. One commonly used method orders the symbols as shown in the above image: grid, grid, X, X. Another commonly used system orders the symbols as grid, X, grid, X. Another is grid, grid, grid, with each cell having a letter of the alphabet, and the last one having an '&' character. Letters from the first grid have no dot, letters from the second each have one dot, and letters from the third each have two dots. Another variation of this last one is called the Newark Cipher, which instead of dots uses one to three short lines which may be projecting in any length or orientation. This gives the illusion of a larger number of different characters than actually exist.^[18]

Another system, used by the Rosicrucians, used a single grid of nine cells, and 1 to 3 dots in each cell or 'pen'. So ABC would be in the top left pen, followed by DEF and GHI on the first line, then groups of JKL MNO PQR on the second, and STU VWX YZ on the third.^[2][14] When enciphered, the location of the dot in each symbol (left, center, or right), would indicate which letter in that pen was represented.^[1][14] More difficult systems use a non-standard form of the alphabet, such as writing it backwards in the grid, up and down in the columns,^[4] or a completely randomized set of letters.

The Templar cipher is a method claimed to have been used by the Knights Templar. It uses a variant of a Maltese Cross.^[19]

Notes[edit]

1. ^ ^abWrixon, pp. 182–183
2. ^ ^abBarker, p. 40
3. ^ ^abWrixon, p. 27
4. ^ ^abGardner
5. ^Bauer, Friedrich L. 'Encryption Steps: Simple Substitution.' Decrypted Secrets: Methods and Maxims of Cryptology (2007): 43.
6. ^Newby, Peter. 'Maggie Had A Little Pigpen.' Word Ways 24.2 (1991): 13.
7. ^Blavatsky, Helena Petrovna. The Theosophical Glossary. Theosophical Publishing Society, 1892, pg 230
8. ^Mathers, SL MacGregor. The Kabbalah Unveiled. Routledge, 2017, pg 10
9. ^Thompson, Dave. 'Elliptic Curve Cryptography.' (2016)
10. ^MacNulty, W. K. (2006). Freemasonry: symbols, secrets, significance. London: Thames & Hudson, pg 269
11. ^Parrangan, Dwijayanto G., and Theofilus Parrangan. 'New Simple Algorithm for Detecting the Meaning of Pigpen Chiper Boy Scout (“Pramuka”).' International Journal of Signal Processing, Image Processing and Pattern Recognition 6.5 (2013): 305-314.
12. ^Agrippa, Henry Cornelius. 'Three Books of Occult Philosophy, or of.' JF (London, Gregory Moule, 1650) (1997): 14-15.
13. ^Agrippa, Cornelius. 'Three Books of Occult Philosophy', http://www.esotericarchives.com/agrippa/agripp3c.htm#chap30
14. ^ ^abcdPratt, pp. 142–143
15. ^Hynson, Colin. 'Codes and ciphers.' 5 to 7 Educator 2006.14 (2006): v-vi.
16. ^Kahn, 1967, p.~772
17. ^Newton, 1998, p. 113
18. ^Glossary of Cryptography
19. ^McKeown, Trevor W. 'Purported Templars cipher'. freemasonry.bcy.ca. Retrieved 2016-11-07.

Dots And Boxes Game Pigeon Game

References[edit]

• Barker, Wayne G., ed. (1978). The History of Codes and Ciphers in the United States Prior to World War I. Aegean Park Press. ISBN0-89412-026-3.
• Gardner, Martin (1972). Codes, ciphers and secret writing. ISBN0-486-24761-9.
• Kahn, David (1967). The Codebreakers. The Story of Secret Writing. Macmillan.
• Kahn, David (1996). The Codebreakers. The Story of Secret Writing. Scribner. ISBN0-684-83130-9.
• Newton, David E. (1998). 'Freemason's Cipher'. Encyclopedia of Cryptology. ISBN0-87436-772-7.
• Pratt, Fletcher (1939). Secret and Urgent: The story of codes and ciphers. Aegean Park Press. ISBN0-89412-261-4.
• Shulman, David; Weintraub, Joseph (1961). A glossary of cryptography. Crypto Press. p. 44.
• Wrixon, Fred B. (1998). Codes, Ciphers, and other Cryptic & Clandestine Communication. Black Dog & Leventhal Publishers, Inc. ISBN1-57912-040-7.

External links[edit]

• Online Pigpen cipher tool for enciphering small messages.
• Online Pigpen cipher tool for deciphering small messages.
• Deciphering An Ominous Cryptogram on a Manhattan Tomb presents a Pigpen cipher variant

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