Hockey Ice Time Math Problem

Recently, I was involved in a discussion with the coaches of my son’s hockey team surrounding the length of time each player is on the ice for any given game. What spurred the conversation was the absence of one of the six defencemen on the team, leaving the other five players with a bit more playing time.

By the end of the discussion I was left wondering what the outcome would be if teams with six players on defence, made the decision to have one, or two, or three, sit out each game, thereby giving those players in the game more ice time. What impact would this have on the overall season totals of ice time for each player involved? Going into the problem, I knew that when one, or two, players were missing due to illness or injury, it left the remaining players to play more often in a game, getting more ice time, but also feeling like they were more involved in the game. I wasn’t sure how the numbers would turn out, but I put it to the test.

Here are the numbers I was working with:

6 defence on the team

2 defence are on the ice at all times

35 minute game length (10-10-15) [this meant splitting 70 minutes of ice time between the 6 players]

I split the time evenly, since at this level, no adjustments are made for special circumstances and I assumed over the course of the entire season, the amount of ice time would essentially be even between all 6 players

70 games going to be played (this came from how many games were played last season)

Here are the various scenarios:

Scenario 1: All 6 defence are healthy and play all 70 games

Scenario 2: Only 5 defence dress for each game; 1 player sits out each game; this works on a rotating basis with each player sitting out every 6th game (ABCDE – F; ABCDF – E; ABCEF – D; ABDEF – C; ACDEF – B; BCDEF – A; repeat)

Scenario 3: Only 4 defence dress for each game; 2 players sit out each game; this works on a rotating basis with each player sitting out every 3rd game (ABCD – EF; ABEF – CD; CDEF – AB; repeat)

Scenario 4: Only 3 defence dress for each game;  3 players sit out each game; this works on a rotating basis with each player sitting out every 2nd game (ABC – DEF; DEF – ABC; repeat)

So… what total ice time do you get for each player in each scenario?

When I came up with a solution, and when my class (@CaledonEastJELC) tried this problem, we didn’t stray from 35 minute games and 70 games played, but I would love to see solutions which increase game length and/or games played.

I’d love to know the solutions you come up with. Please send any solutions to mrfletcher205@gmail.com or on Twitter to @HeyMrFletcher