#### Introduction

In a recent post Tim Jenkins covered the 1987 Kraft “Home Plate Heroes” baseball card set. One detail of the set he addressed was the multiple player combo variations. Specifically, each of the 48 players in the set was paired with five other players. The ex-mathematician in me found this aspect of the set intriguing enough to (depending on your taste) promise or threaten to make it the subject of my next post, in hopes that some elegant mathematical structure would underlie the 120-panel master set.

#### A quick look at some of tim’s cards

Borrowing an example and some graphics from Tim’s original post, here is Don Mattingly paired with Fernando Valenzuela and Steve Garvey. Other Mattingly panels include Hubie Brooks, Mike Scott, and Mike Schmidt.

Since my analysis is more about numbering than the players themselves, here are the card numbers of Mattingly’s five partners.

- #4 – Mike Scott
- #20 – Steve Garvey
- #30 – Mike Schmidt
- #32 – Fernando Valenzuela
- #42 – Hubie Brooks

Were you to look for a single rule or relationship behind these numbers, you’d probably come up empty, even as you might jot down a handful of properties.

- All the Mattingly partners have even numbers
- There are two instances where the numbers are 10 apart (20/30 and 32/42). Furthermore, because the set ends at 48, even the 42 and 4 can be considered 10 apart, modulo the size of the set. (Don’t panic if that last bit means nothing to you.)
- Finally, as Tim already noted in his article, one of the numbers (30) is one greater than the Mattingly card (29).

A fair question at this point is whether these properties are unique to the Mattingly panels or hold more generally across the set.

#### Are they steady, eddie?

The first card in the Kraft set is this Eddie Murray card.

Without too much trouble, I was able to find the five players paired with Murray on uncut panels:

- #2 – Dale Murphy
- #4 – Mike Scott
- #22 – John Tudor
- #24 – Von Hayes
- #48 – Nolan Ryan

Returning to our Mattingly observations we see the first and third still hold, but the “10 apart” property no longer applies to any of the card numbers. Were we to quit now, we might conclude the logic to the set was simply this:

- Odd numbers on the left, even numbers on the right
- For any player on the left, one of five panels should include the next number in the set
- The other four panels should include (perhaps) randomly chosen even numbered cards

For most sets of baseball cards, it would not surprise me at all (even if it would disappoint me) to learn that little thought went into a set’s checklist or numbering. After all, chaos is usually easier than order. However, in the case of the Kraft set, I found such an outcome difficult to accept, simply because it is actually a hard and time-consuming exercise to arrive at five panels per player through randomness or luck.

#### What would Albert Einstein do?

This quote, probably never uttered by Albert Einstein, comes to mind.

At least in theory, developing a rule to generate a 120-panel master set with five pairings per player should take far less time than throwing darts and attempting a checklist willy-nilly.

#### Building a checklist

If I were to make it any farther with this set I knew I would need a complete checklist of all 120 panels. Failing to find one anywhere online, I was able to build my own, which (drum roll please…) I’ll be sharing for your research or collecting pleasure at the end of this article. (And yes, I did see that the PSA Card Facts page for this set lists panels; however, it had numerous problems that made it unreliable as a sole source. For instance, it lists 129 panels, frequently swaps the left/right pairings, picks a bad year to not tell us which Davis, and fails to number any of the cards.)

Once my checklist was built and sorted appropriately, new patterns were evident just from looking at the first three entries. It turns out the key to finding patterns wasn’t to look across but look down.

Where panel pairings looked nearly random previously, there were now simple patterns like 2, 4, 6, … and 48, 46, 44, … that were clearly intentional and seemed to decode the entire master set checklist.

#### But…

At first glance, such a numbering scheme appears to be a slick and elegant way to generate the master set. There is only one problem. Because the numbering in the columns goes both forward (Var 1, Var 2, Var 4) and backward (Var 3, Var 5), there is risk of the different columns ultimately “crashing” into each other and landing on the same number at the same time. (Crashing is not actually guaranteed but also depends on the starting point of each pattern. In a simpler non-card example, the patterns 1, 2, 3, … and 3, 2, 1, … both have the same middle number, but the patterns 1, 2, 3, 4, … and 4, 3, 2, 1, … have no terms in common.)

For better or worse, we indeed encounter a collision when we hit Ozzie Guillen’s card #11. Without further adjustment, this would translate into only four distinct Guillen variations, with panel 11/12 (Guillen/Pena) double-printed.

Continuing each pattern down the rest of the checklist, a total of five collisions result.

One option for Kraft would have been to declare “good enough” and apply this scheme unaltered for the master set. The result would be 115 different panels instead of 120, with five of them double-printed. In the grand scheme of things I think most collectors would have either not noticed or not complained. After all, how many collectors were really looking to buy 120 boxes of macaroni in the first place?

Of course that’s not what happened. Kraft did in fact issue 120 distinct panels. To do so, Kraft had to resolve each of the five collisions without creating new ones. Their primary strategy for addressing collisions was a clever one: simply to swap unwanted duplicates with the next number in the sequence. This is exactly what was done in the first instance, where Guillen’s second Pena (12) was traded for the Eric Davis (10) that would have gone with Harold Baines.

Such a strategy, if exploited fully, could have been used to resolve all but one conflict on the checklist. The lone exception would have been the #23 Yount (23) row, where flipping either 26 would cause duplication in the Hrbek (25) row.

Still, why let perfect get in the way of good? Perhaps a 119-panel master set would have set off a macaroni buying frenzy among collectors determined to find the modern day equivalent of a 1933 Goudey Nap Lajoie! (Yes, I’m kidding. Based on the set’s packaging, shoppers could easily tell what cards were on the boxes just by flipping them over.)

Yet another solution available, largely in the spirit of prior interventions, would have been to swap Yount’s second 26 (var 5) with the number immediately above it. Ultimately, however, Kraft took a different path with Yount as well as a couple of the earlier checklist crashes.

#### Unleash the cheese!

Rather than fully exploit down-flip or up-flip strategies to resolve all five conflicts on the checklist, Kraft applied such flips only three times (rows 11/13, 35/37, 37/29). For the #13 Baines and #23 Yount rows, the adjustments seemed more oddball. As the yellow cells in the table show, Kraft replaced Baines’ extra 36 with a 22 and Yount’s extra 26 with a 2! Huh?!

In reality the mysterious choices of 22 and 2 aren’t so mysterious. Take a look at the very end of the checklist, and you’ll see that the final two entries should have been…you guessed it…22 and 2! It’s not terribly elegant, but neither is it random. And of course it worked!

The result, for the dining and collecting pleasure, of many of us was a fun 48-card set that included five different ways to collect each player (my favorite: Kirk Gibson/Steve Garvey), based on a handful of simple rules for checklist design and two resolution techniques for the conflicts these rules produced.

#### Extra for experts

As a final note, perhaps for someone else to tackle, I mentioned a few times that crashes in the checklist would have led to double-printing of some panels and the absence of others. In fact, it’s still possible that double-printing did occur and that the fixes were added as extras rather than replacements. Population counts for panels are currently too low to get much of a sense of things, but then again this was 1987. What does double-printing even mean when you print a billion of everything? đ

#### Appendix: Master set checklist

I have posted a complete checklist of panels as a Google Sheet. Here is the link. And if you have an extra Kirk Gibson/Steve Garvey panel somewhere, give me a holler!