The NHL recently announced an expansion team in Las Vegas bringing the league to 31 teams. They also announced their scheduling matrix. It won't work.
Each Pacific team plays 28 games, 14 home, 14 away. The 29th game is a home game against a randomly selected team who is also at home when they play that game (as an example, CGY-EDM at both, VAN-SJS at both, LAK-ANA at both, LSV-ARZ at both). Pacific Division teams will play 11 home games and 10 away games against the Central Division, as well as 16 home and 16 away games against the Eastern Conference. So each Pacific team will play 42 (41½) home games and 40 (40½) away games.
Central Division teams play 13-13 in the division, 12-12 against the Pacific, and 16-16 against the East for a total of 41-41. Eastern conference teams play 14-14 in the division, 12-12 against the other division in their conference, and 15-15 against the west, for a total of 41-41.
Let's look a little closer at games between the Central Division and the Pacific Division. Of these 168 games, 7 x 12 = 84 are home games for the Central, 7 x 12 = 84 are away games for the Central, 8 x 11 = 88 are home games for the Pacific, and 8 x 10 = 80 are away games for the Pacific. So four of these games have to be home games for both teams.
I may update or post again when I have a solution.
UPDATE:
I have found a solution that gives each Western team 25 home games and 25 away games.
The subdivisions are arbitrary and are only used for discussion.
Pacific subdivision A: Kings, Ducks, Sharks, Canucks ; Pacific Subdivision B: Flames, Oilers, Coyotes, Vegas.
Central subdivision aa: Stars, Blues, BlackHawks, Wild ; Central Subdivision bb: Predators, Coyotes, Jets.
Pacific A: 2 home games against Pacific subdivision A. 2 home games against Pacific subdivision B + 1 additional home game against one team from PS B.
1 home game against CS aa, 2 home games against CS bb.
2 away games against every Pacific team, 2 away games against CS aa, and one home game against CS bb.
Division: 15 home, 14 away; conference crossover: 10 home, 11 away.
Pacific B: 2 home games against every Pacific team. 2 home games against CS aa and 1 against CS bb.
2 away games against PS B, 2 away games agains PS A + 1 additional away game against one team from PS A. 1 Away game against CS aa and 2 away games at CS bb.
Division: 14 home, 15 away; conference crossover: 11 home, 10 away.
Central aa: 2 Home games against every central aa team + 1 additional home game against one CS aa team. 2 Home games against every Central bb team. 2 home games against every PS A team, and 1 home game against every PS B team.
2 Away games against every central aa team + 1 additional away game against one CS aa team. 2 away games against every CS bb team. 1 Away game against every PS A team, and 2 away games against every PS B team.
Division: 13 home, 13 away; conference crossover: 12 home, 12 away.
Central bb: 2 Home games against every central aa team. 2 Home games against every Central bb team + 1 additional home game against one CS bb team. 1 home games against every PS A team, and 2 home game against every PS B team.
2 Away games against every central aa team. 2 away games against every CS bb team + 1 additional away game against one CS bb team. 2 Away game against every PS A team, and 1 away games against every PS B team.
Division: 13 home, 13 away; conference crossover: 12 home, 12 away.
This may be what they meant, but they did not say it very well.