The recreation facilities on Starfleet vessels were typically equipped with several board games of Terran or alien origin. These games were quite popular among Starfleet crews as an off-duty pastime. There were two-dimensional as well as three-dimensional board games. 2D games were played on a single, flat game board, whereas 3D games usually featured several flat game boards on different planes.
2D board games[]
The recreation room on Enterprise NX-01 was equipped with a standard Terran chess set. (ENT: "Observer Effect") Ensign Jean-Luc Picard also kept a chess set with large, frosted glass pieces in his quarters at Starbase Earhart. (TNG: "Tapestry")
Trip Tucker kept the equipment to play Go in his quarters aboard Enterprise NX-01. (ENT: "Cogenitor")
Kadis-kot was frequently played in the mess hall of the USS Voyager. (VOY: "Endgame", et al.) When a hologram of Reginald Barclay apologized to The Doctor for cancelling their game of golf, saying Seven of Nine was going to teach him kadis-kot, The Doctor was miffed, saying he lent him his mobile emitter to conduct business, not play board games. (VOY: "Inside Man")
In the late 24th century, the Romulans on Vashti played a three-person game that involved a board with pieces and pixmit cards. (PIC: "Absolute Candor")
According to the call sheet for the seventh day of filming of the Star Trek: Deep Space Nine third season episode "Meridian", Thursday 29 September 1994, board games were part of the set decoration of the settlement on Meridian though in the final episode no board game can be seen.
3D board games[]
A very popular 3D board game in Starfleet was three-dimensional chess. 3D chess sets were present in the recreational facilities aboard the USS Enterprise, the USS Defiant, and the USS Enterprise-D, there most notably in Ten Forward. (TOS: "Where No Man Has Gone Before"; ENT: "In a Mirror, Darkly"; TNG: "Q Who")
The recreation room on the Constitution-class Enterprise was also equipped with a 3D board game consisting of four vertical acrylic rods in a quadratic arrangement, where each rod supported three 2×2 game boards which closely resembled the moving boards on a 3D chess set, so that there were twelve boards with a total of forty-eight squares (twenty-four white and twenty-four transparent). The game was played with red and black checkers pieces, eight of each color. (TOS: "The Naked Time"; DS9: "Trials and Tribble-ations")
This prop was an actual marketed game called "Space Checkers".
Ten forward on the Enterprise-D was equipped with a three-leveled board game. The game set consisted of two chess-like 7×7 checkered boards on the upper two levels, and a lower board of the same overall size, but closely resembling a Nine Men Morris board. The three levels were arranged stair-like. The game was played with silver and gold game pieces which were all alike. (TNG: "Pen Pals") Variations of this game were also present in the guest quarters aboard the ship. (TNG: "Symbiosis")
The Ferengi Nibor had this game aboard the Krayton, where it was explicitly referred to as chess, with the pieces having the same names (queen, knight, etc.) and using the same terms (check, checkmate) as other forms of chess. Nibor and William T. Riker played a game with this board when Riker was being held prisoner, as a rematch to a previous chess game that Riker won on the Enterprise. (TNG: "Ménage à Troi")
Ten Forward also provided the facilities to play Strategema, a 3D board game which was played with a holographic projection of three game boards instead of a solid game set. (TNG: "Peak Performance")