A harmonic take on Immaculate Grid

Immaculate Grid is a game for baseball trivia nerds. Put out by Baseball Reference, it asks you to name 9 different players in a grid, each having two characteristics, such as “Played for the Rockies” and “Hit over 300 career HR”, which you would promptly miss because noone has any idea what the hell is going on with the Rockies. One of my solved grids is below. The percentages at the upper right of each players image are the fraction of people that used that player. The sum of the percentages is your score, the lower the better. So there’s an incentive to try to think of strange players that meet each category, kind of a reverse Keynesian beauty pageant, if you will.

Most puzzles will have at least one row and one column which are individual teams, which means that you need to find a player who was (as in the above example) on both the Guardians and the Yankees. If you miss one, or you get a high score on a player, you can click through and Baseball Reference will show you everyone who met the criterion, with a list of the Top 5 players ranked by career WAR, or Wins Above Replacement. WAR tries to put a number to how useful a player was relative to a player that basically any team could get at any time, a “replacement-level player”.¹ Notionally, the Top 5 ranking is by total career WAR, but Knucksie doesn’t show up on the Guardians/Yankees list in spite of having more WAR than everyone on it.²

What this is lacking is a ranking of which players might be the most readily associated with both teams. Let’s stick with Guardians/Yankees for now. C.C. Sabathia (who I used above for a different combination, Brewers/Yankees) is fairly well-associated with both teams, coming up with Cleveland and winning his Cy Young there, then playing the second half of his career in the Bronx, and winning the 2009 Series in pinstripes. By contrast, Gaylord Perry was a great pitcher overall who also won a Cy Young in Cleveland³, but only played a forgettable half-season in New York. Both are ranked in the Top 5 for the Guardians-Yankees franchise pair, but Sabathia is much more associated with both teams.

There is a great way to assess this, using a type of average called the harmonic mean. Baseball already uses the harmonic mean for a different statistic called the Power-Speed Number (PSN). To compute it,

where HR is the number of home runs hit by the player, and SB is the number of stolen bases. The PSN can be computed over a career ⁴ or over a season⁵. Why it’s called the harmonic mean is not germane to the present discussion, but what is interesting about it is the way it biases toward the lower number without completely ignoring the higher one.

You can probably see where this is going. For each combination of two teams, it is possible to compute, a similar number. For the Guardians and the Yankees, we have the Guardians-Yankees number. For the Tigers and the Reds, the Tigers-Reds number, etc. There are 435 of these numbers: one for each combination of 30 teams, not counting the trivial combinations where the team is paired with itself. Not all of them will be interesting, but some are! Since we know the characters already, we start with the Guardians-Yankees number:

where is the WAR a player totaled for the Guardians, and is the same for the Yankees. The top two players are C.C. Sabathia with a CLE/NYY of 28.4, and Graig Nettles with 25.5. Both of these players are in the Top 5 by total career WAR among players that spent time with both franchises, but this is not always the case.

For example, consider the Tigers/Red Sox number, DET/BOS. The Top 5 listed for this franchise pair are Dizzy Trout, Bobby Veach, Fred Lynn, Johnny Damon, and Dick McAuliffe. The highest DET/BOS scores are Howard Ehmke (15.8) and Dutch Leonard (14.2), two pitchers that ended their careers before WWII.⁶ Number 3 for this franchise pair is still active: J.D. Martinez scores a 13.5 on the back of an exceptionally balanced 13.2/13.7 WAR for Detroit and Boston, respectively. He’s no longer active with either team, but an additional 1.7 WAR for the Tigers would push him over Leonard.

What does this average mean? I think it’s a way of measuring the following: if you paid some attention to both teams during a player’s career, who would jump out at you the most? It obviously can’t account for recency, or extremely short notable stints with a team, like Doyle Alexander on the Tigers in 1987, or Sabathia’s preposterous half-season on the Brewers in 2008.

I haven’t computed every combination yet, so I’m planning to plug away at these and do little writeups as I get them done.

(Featured image original from Cooperstown Expert, modified by Aren, questions best directed to Bluesky)

• Bluesky

Completed to date

Guardians-Yankees: C.C. Sabathia (28.4) and Graig Nettles (25.5).

Tigers-Red Sox: Howard Ehmke (15.8) and Dutch Leonard (14.2). J.D. Martinez (13.5) is still active, and passes Dutch with another 1.7 WAR for Detroit.

Tigers-Reds: Sam Crawford (11.6) and Harry Heilmann (8.3). Both are in the Top 5 for the franchise pair, and both are in the Hall of Fame.

Diamondbacks-Rangers: Doug Davis (5.7) and Rick Helling (3.5). Both are in the Top 5 for the franchise pair. Max Scherzer is still active at time of writing, and would pass Helling with another 4.0 WAR for Texas. He played for Texas in 2024, but is a free agent at time of writing.

1. More details for the interested.
2. Ah, the knuckleball.
3. Ah, the spitball.
4. Highest career: Barry Bonds, 613.9
5. Shohei Ohtani, 56.4 in 2024, also the highest season mark of all time.
6. Full disclosure: Ehmke looks to have slightly more career WAR than Dick McAuliffe and should probably be in the Top 5 despite not being listed. I sympathize, especially if these Top 5 lists are done manually. I’ve yet to automate things at time of writing, and I could miss someone.