Welcome to The STEM Sessions Podcast. I am your host, Jarl Cody.

Played hockey since high school

- Consistently in an adult league for over 20 years
- Same group of guys for half of that

My team currently has 12 skaters on the roster

- We’re all busy guys, and on any given week it seems 2-5 of us miss the game – life happens
- Despite getting old, we’re very used to playing with seven or eight guys
- Most of us have the experience to play every position to some level of success
- So we just roll with whatever line combinations are necessary that week

However, about once a season, or maybe every other season, we hit the ice with seven forwards, and that flexibility and sensibility falls apart

- Bench descends into chaos
- No one can figure out who gets on the ice next
- Guys on my team aren’t dumb – we have engineers, guys in tech and finance and medicine, business owners, teachers, writers
- Still spend the entire first period floundering when we have seven forwards and have to run a rotator
- Once or twice I’ve been so frustrated I moved myself to forward and played as the rotator just to stop the discussion

Finally decided enough’s enough

- Determine the optimal way to run the rotator
- figured there had to be an easy mathematical way to find it

This is The STEM Sessions Podcast Episode 22 – The Seventh Forward

ice hockey team has six players on the ice

- One goalie, two defensemen (called a defensive pair), and three forwards (a center, right winger, and left winger together called a line)
- For purposes of this discussion, we’ll focus on the forwards

During game, NHL will dress 20 players

- Two goalies (starter and back up)
- Six defensemen (three pairs)
- 12 forwards divided into four lines
- Top two forward lines and top two defensive pairs get more playing time

In adult recreation leagues, or beer leagues, rosters are not that large, and playing time should be equal

- Line combos need to be flexible depending on who makes it to the game
- Six or seven skaters means everybody plays every position and you can optimize on the ice
- Eight or nine skaters means you have three defensemen and the rest as forwards
- Ten skaters means you have two forward lines and two defensive pairs
- 12 and 13 skaters means four defensemen and the rest as forwards, though no one wants to play with that many skaters

Eleven skaters is really the only number that feels weird

- Three defense, eight forwards is unbalanced in terms of playing time
- While four defense, seven forwards is better, arranging the forwards is complicated

What to do with the seventh forward?

- Some teams will run three left wings, two centers, and two right wings, or similar arrangements
- But that isn’t fair to the skaters playing the position with three
- So most teams will run a rotator
- one skater moves through the forward lines swapping in for a different player each shift, sitting out every fourth
- Theoretically, minimizes discrepancy in playing time among forwards
- But how do we determine the best way to run the rotator through the lines?

First approach to find the best way was to look at playing time

- number is minutes in a period is P
- minutes in a game 3P
- Ignoring penalty kills, three forwards on the ice means a total of 9P minutes to share among all forward
- If two even lines of forwards, each gets 9P divided by six or 1.5P of playing time
- If rotator, each forward should get 9P divided seven or 1.28P of playing time

That makes sense and is obvious in hindsight, but does nothing to tell me how to achieve that balance

- so next i looked at number of shifts because that’s more controllable during the game
- Takes 4 shifts for rotator to move through one line – one shift at each position, then a shift sitting
- During that series, second line has three shifts
- Therefore, 7 shifts total are required to make a complete rotation
- During that rotation, all seven forwards each have three shifts
- Ideal number of shifts is a multiple of 7

To determine shift length, let shift length be S

- Total minutes in game is 3P (as discussed above)
- Shift length is therefore 3P divided by 7N, where 7N is the 7 shifts times a whole number multiplier
- Or S equals 0.43P divided by N

In my case, we play 14 minutes periods

- 7 shifts would be six minutes each
- 14 shifts would be three minutes each
- 21 shifts would be two minutes each
- 28 shifts would be 1.5 minutes
- 35 shifts would be 1.2 minutes
- And so on

That tells us the number of shifts to target, but still doesn’t tell us “how” to rotate

- So being the engineer I am, and with an elegant mathematical solution far from my grasp, decided to brute force it
- Pulled up excel
- Line 1 designated as L1, C1, R1
- Line 2 as L2, C2, R2
- Rotator is X

Constructed number scenarios

- varying number of shifts per period
- Important to note I used shifts per period instead of shifts per game
- This means some multiples of 7 shifts are not possible
- But having shifts not carry over from one period to the next is often more realistic in beer league
- Also compared sitting the rotator on the first shift and playing him on the first shift
- And should the rotator switch lines after each rotation or wait until the half-way point of the game

Here is what I concluded

Rule 1 – rotator should switch lines after each rotation

- When alternating lines, 7 shifts per period and 14 shifts per period results in 9 and 18 shifts for each forward respectively
- Comparing those shift scenarios to their counterparts when the rotator switches lines mid-game, rotator always loses a shift
- Other shifts per period scenarios see similar, if not more severe, losses for the rotator

Rule 2 – the rotator shall start in Line 1 and shall be on ice first shift

- Less important than first rule
- In 7 shifts and 14 shifts per period, forwards will have an equal number of shifts regardless if the rotator plays or sits to start
- But in other scenarios, rotator always at a deficit if he doesn’t start
- Of course this means, if the rotator isn’t at a deficit in shifts another forward is, but I think the rotator should be rewarded for keeping it all running smoothly

Rule 3 – Rotator should be strongest forward

- If you follow the two rules above, the rotator will have as many shifts as the other forwards, and even one more in some scenarios
- Use a strong forward at rotator to maximize effectiveness
- Contrary to what is typically done, so perhaps it’s more of a guideline than rule

Most complaints about rotator are due to imbalanced playing time

- Though I think we tend to exaggerate the deficit in our heads
- Regardless of how you run the rotator, you’re unlikely to end the game with equal shifts just because of the game itself
- Even if you do everything perfectly, penalty kills or getting trapped in your zone will always leave someone short-shifted

So perhaps the most important takeaway is short shifts minimize playing time imbalances

- In some scenarios I created, there was a three shift difference among forwards
- If you’re taking two minute shifts, that’s a difference of six minutes in playing time
- If you’re taking one minute shifts, the imbalance is cut in half
- And if you’re skating hard, I promise you won’t notice those minutes

Unfortunately, keeping shifts short is not a skill practiced by most beer league teams

- So perhaps we are doomed to forever be tormented by the rotator

Thank you for listening to The STEM Sessions Podcast.

This episode was researched, written, and produced by Jarl Cody.

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