A professional game consists of three periods of twenty minutes each, the clock running only when the puck is in play. The teams change ends for the second period, again for the third period, and again at the start of each overtime played. Recreational leagues and children's leagues often play shorter games, generally with three shorter periods of play.
Various procedures are used if a game is tied. In tournament play, as well as in the NHL playoffs, North Americans favor sudden death overtime, in which the teams continue to play twenty minute periods until a goal is scored. Up until the 1999-2000 season regular season NHL games were settled with a single five minute sudden death period with five players (plus a goalie) per side, with both teams awarded one point in the standings in the event of a tie. With a goal, the winning team would be awarded two points and the losing team none (just as if they had lost in regulation).
From 1999-2000 until 2003-04, the National Hockey League decided ties by playing a single five minute sudden death overtime period with each team having four players (plus a goalie) per side to "open-up" the game. In the event of a tie, each team would still receive one point in the standings but in the event of a victory the winning team would be awarded two points in the standings and the losing team one point. The idea was to discourage teams from playing for a tie since previously some teams might have preferred a tie and 1 point to risking a loss and zero points. The only exception to this rule is if a team opts to pull their goalie in exchange for an extra skater during overtime and is subsequently scored upon (an 'Empty Net' goal), in which case the losing team receives no points for the overtime loss.
International play and several North American professional leagues, including the NHL (in the regular season), now use an overtime period identical to that from 99-00 - 03-04 followed by a penalty shootout. If the score remains tied after an extra overtime period, the subsequent shootout consists of three players from each team taking penalty shots. After these six total shots, the team with the most goals is awarded the victory. If the score is still tied, the shootout then proceeds to a sudden death format. Regardless of the number of goals scored during the shootout by either team, the final score recorded will award the winning team one more goal than the score at the end of regulation time. In the NHL if a game is decided by a shootout the winning team is awarded two points in the standings and the losing team is awarded one point. Ties no longer occur in the NHL.
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I think you're referring to NBA and College Basketball games. NBA games consist of four, 12-minute quarters while college Basketball games consist of two, 20-minute halves. I'm not sure what the reasons are other than the level of talent of each league.
While playing Ice Hockey, the ice surface needs to be kept smooth. Hence in order to avoid resurfacing of ice, the interval is fixed to be 20 minutes.