The Black Path Game (also known by various other names, such as Brick) is a two-player
board game
Board games are tabletop games that typically use . These pieces are moved or placed on a pre-marked board (playing surface) and often include elements of table, card, role-playing, and miniatures games as well.
Many board games feature a comp ...
described and analysed in ''
Winning Ways for your Mathematical Plays''. It was invented by Larry Black in 1960.
[.]
It has also been reported that a game known as "Black" or "Black's Game" was invented in 1960 by William L. Black. This "William L. Black" (possibly known as "Larry") was at that time an undergraduate at the
Massachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
, investigating
''Hex'' and ''
Bridg-It
The Shannon switching game is a connection game for two players, invented by American mathematician and electrical engineer Claude Shannon, the "father of information theory" some time before 1951. Two players take turns coloring the edges of an a ...
'', two games based on the challenge to create a connected "chain" of counters that link opposite sides of a game board. The creative outcome of Black's research was a new topological game that his friends (perhaps unimaginatively) called ''Black''. The game was introduced to the public by
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
in his October 1963 "
Mathematical Games column
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, through June 1986, Gardner wrote 9 more columns, ...
" in ''
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
''.
Rules
The Black Path Game is played on a board ruled into squares. One edge on the boundary of the board is designated to be the start of the path. After the first move, the players extend the path away from the starting edge by alternately filling the adjacent square at the end of the current path with one of three configurations shown below.
Any square that is not empty is filled with one of the following configurations that contains two paths linking two sides:
Image:Square (02) 12-34.svg, T1 tile, with paths connecting adjacent sides (top and left, and bottom and right)
Image:Square (02) 14-23.svg, T2 tile, with paths connecting adjacent sides (top and right, and bottom and left)
Image:Square (20) 13-24.svg, T3 tile, with paths connecting opposite sides (top and bottom, and left and right)
These tiles are the three ways to join the sides of the square in pairs. The first two are
the tiles of the
Truchet tiling.
The path may return to a previously filled square and follow the yet-unused segment on that square. The player who first causes the path to run back into the edge of the board loses the game.
Strategy
As outlined in the example games provided, the player who routes the path into a corner of the board will win the game, as the other player will have no choice but to run the path into the edge of the board.
[
The first player has a winning strategy on any rectangular board with at least one side-length even so there are an even number of squares in total. Imagine the board covered with rectangular (2×1 unit-size) ]domino
Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also ca ...
es. If the first player always plays so the end of the path falls on the middle of one of the dominoes, that player will win. This strategy was discovered by Black's friend Elwyn R. Berlekamp
Elwyn Ralph Berlekamp (September 6, 1940 – April 9, 2019) was a professor of mathematics and computer science at the University of California, Berkeley.Contributors, ''IEEE Transactions on Information Theory'' 42, #3 (May 1996), p. 1048. DO10.1 ...
,[ who subsequently described it in his book.][
If both sides of the board are odd, the second player can instead win by using a similar domino tiling strategy, including every square but the one containing the first player's first move.]
Logic
The domino tiling strategy works by making the losing player end the path on the edge of a new domino; by continuing the path on the new domino, the winning player will eventually force the losing player to the edge or a corner.[ Player 2 can win on an even-celled board; first consider the board completely covered with 2×1 dominoes except for the upper left and lower right corners. If Player 2 forces Player 1 to move in 2 the second cell of the main diagonal, regardless of Player 1's move in 2 the unused path in 2will connect two squares which can be regarded as the two squares of a "split domino" that Player 2 can use, and the remaining tiles (save the lower-right corner) can be covered in dominoes.][
Refer to the three examples below, illustrating the "split domino" that results from the third move.
]
Examples
Consider the example game shown at right on a 4×4 grid, where the moves have been:
# 1T2
# 1T2
# 2T3
# 3T2
# 3T1
# 2T2
# 2T1
# 3T2
According to the rules, the next move by Player 1 (odd-numbered turns) must be in space 4to continue the path. If Player 1 makes the move 4T3 that will result in an instant loss, since this tile will link the path to the bottom edge. Playing 4T1
will result in a win, as Player 2's following move is placed in the corner 4and Player 3 will lose regardless of the piece played. Playing 4T2
results in an eventual loss for Player 1; comparison to the 4×4 domino-tiled blank board shows that Player 1 making the move 4T1 puts the path into the middle of the domino, while 4T2 puts the path onto the edge of the domino, and Player 2 can maneuver Player 1 into making the last move.
Gardner describes a second example game, where the moves have been:
# 1T3
# 2T2
# 2T3
# 2T1
# 1T2
# 1T1
# 3T2
# 3T1
# 2T1
In this second example, Player 1 has maneuvered the path into the corner space 1 which results in a win regardless of the move made by Player 2.[ After the third move (by Player 2) in 2T3, the split domino exists in cells 1and 3 However, the fifth move by Player 2 1T2 brought the path to the edge of the split domino, which Player 1 took advantage of with the sixth move 1T1, playing to the middle of the split domino.
]
See also
* Tantrix
''Tantrix'' is a hexagonal tile-based abstract game invented by Mike McManaway from New Zealand. Each of the 56 different tiles in the set contains three lines, going from one edge of the tile to another. No two lines on a tile have the same ...
, which uses hex serpentiles Serpentiles is the name coined by Kurt N. Van Ness for the hexagonal tiles used in various edge-matching puzzle connection abstract strategy games, such as Psyche-Paths, Kaliko, and Tantrix. For each tile, one to three colors are used to draw path ...
with three paths and six entry points; each path has a different color
* Trax, which uses a similar set of square tiles with two paths and four entry points; each path also has a different color
* Tsuro
Tsuro is a tile-based board game designed by Tom McMurchie, originally published by WizKids and now published by Calliope Games.
Tsuro is a board game for two to eight players. To play, players compete to have the last playing piece remaining o ...
, which uses square tiles with four paths and eight entry points
References
{{reflist
Board games introduced in 1960
Abstract strategy games
Mathematical games
Connection games