#### DR K. Ten Most Wanted

One of the coolest cards on my Dr. K “Ten Most Wanted” list as of last week was his 1991 Topps Cracker Jack 4-in-1 panel. (And please get in touch if you can help with any of the others!)

As a student of the master, I try hard not to pay more than a buck or two for any of the cards in my Gooden collection, but I finally bit the bullet and threw real money at this Cracker Jack panel.

I was particularly swayed by Doc’s co-stars, three of my favorite players of the era: Tony Gwynn, Eric Davis, and Ken Griffey, Jr.

#### Variations

As is the case with many panel-based sets there were in fact multiple panel variations featuring different player combinations.

It didn’t seem to take co-chair Nick more than a minute to figure out that there was something very not random about these player combinations. And sure enough, they all point to a larger sheet with exactly this neighborhood around Gooden.

Before going too much further into sheets or panels I should pause and provide more background on the individual cards themselves.

#### Like the regular cards but smaller

The 1991 Topps flagship set had the era-typical 792 cards and nearly as many spin-offs as Happy Days. Think Topps Desert Shield for example. Two of them featured tiny cards: the 792-card Micro set and the 72-card Cracker Jack set that itself was divided into a Series One and Series Two, each numbered 1-36. The Cracker Jacks are occasionally referred to as Micro cards with different backs, but this isn’t quite correct since card size differs as well.

Speaking of backs, here is what the Cracker Jack card backs look like. Series One is on the left, and Series two is on the right.

Apart from the different colors used, you’ll notice the second card also has “2nd Series” in the top left corner whereas the first card does not indicate any series at all. This suggests Cracker Jack (or Borden for you corporate types) originally planned only a single series of 36 cards and then added another due to the popularity of the initial offering. The star power of the first series checklist vs the second also seems to point in this direction.

#### Population report

The popularity of the promotion is borne out by a highly informative (if occasionally inaccurate) business article on the Borden-Topps partnership that even includes sales figures for each series:

- Series One: 75 million boxes
- Series Two: 60 million boxes

Based on the size of the set, this translates into about 2.1 million of each first series card and 1.7 million of each second series card. Back in 1991 we would regard this as “limited edition” with the second serious “extra scarce.”

#### SOURCE OF 4-in-1 PANELS

Though I’m sure the truth is out there, I can find no definitive documentation on the source of the 4-in-1 panels. Theories I’ve run across include–

- Distributed to dealers or show attendees as promos
- “Handmade” from full sheets of 1991 Topps Cracker Jack
- Part of a mail-in offer

I’ll regard the first two as most viable at the moment. Mail-in seems less likely since Cracker Jack already had an offer in place for mini-binders.

I’ll offer one other very weak clue that works against both the mail-in and promo theories: the size of the borders.

On the left is an actual 4-in-1 panel. You’ll notice the interior borders are about twice the thickness of the outside borders. The card really doesn’t look bad, but I would think if you were designing the card intentionally as a 4-in-1 you might even out the borders to achieve a cleaner look. (And yes, I did say “very weak” clue.)

Regardless, the most fun option is to assume the four-in-one panels were produced on the secondary market from uncut sheets. Of course when I say “fun,” I don’t mean “normal people fun.” I mean mathematically fun!

#### Anyway you slice it

For the moment let’s assume that uncut sheets simply cycle through the 36 cards in the set over and over again in the same order until the end of the sheet is reached. Let’s also assume that our sheet cutter doesn’t just cut any old 2 x 2 array from the sheet but proceeds methodically from the beginning of the sheet and works his way left to right, up to down. Finally, let’s assume cards are sequenced in numerical order.

Here is what this might look like for a sheet that is 18 cards across and 10 cards down the side. What you’ll notice, as in the highlighted blocks of 5-6-23-24 panels, is that there are no variations produced by the method. If card 5 were Dwight Gooden for example, he would always have the same three co-stars.

Without breaking out any fancy number theory, you might correctly surmise that the rather ho-hum result has something to do with the sheet repeating every two rows, which only happens because we conveniently matched the number of columns to exactly half the set length.

Now let’s run the same cut against a sheet that’s only 16 cards across. It stands to reason that at least something more interesting should happen here. Sure enough, the new configuration produces two Gooden panels: 5-6-21-22 and 25-26-5-6.

While two variations is nice, it’s not four. If you think about it, one of the things limiting our variations here is that we always see 6 paired with 5, which is another way of saying we never see 4 paired with 5. (Alternatively, we can say that 5 is always on the left and never on the right.)

So long as we have an even number of columns, this sort of thing will always be true. Therefore, our only hope at hitting four variations would seem to be having an odd number of columns. The example below uses 15 and sure enough pairs 4 with 5 some of the time. Unfortunately, we still failed to reach the theoretical maximum of four variations.

Is it possible we just got unlucky with our odd number? Maybe so. Let’s try this one more time and go with lucky 13.

Alas, we remain stuck at only two variations. Is it possible that there’s simply no solution involving methodical cuts and that achieving more variations will require something more like this?

The good news is we have our four variations. The bad news is we end up with a lot more waste.

#### Is there a Methodical solution?

Let’s take one more quick look at the actual Gooden panels that started this post. In case you missed it the first time, notice that Doc finds himself in a different quadrant on each of the panels. For example, he’s in the upper left position on the Gooden-Boggs-Griffey-McGwire panel, and he’s in the upper right position on the Gwynn-Gooden-Davis-Griffey panel.

Under our assumptions about how uncut sheets should be constructed, the problem of a methodical cut yielding four variations is equivalent to the problem of a methodical cut putting Doc in each of the four panel quadrants.

Looking back at our earlier examples, we can see that our first time through Doc was only in the upper left, which is why we had only a single variation. The next time, Doc was upper left and lower left only. Finally, in our last two methodical efforts, Doc was only upper left or lower right.

What would our sheet need to look like for Doc to end up in all four panel quadrants? We can attempt to solve the problem by examining one quadrant at a time, starting with the upper left quadrant.

For card 5 to end up in an upper left position, we need two things to happen.

- 5 needs to be in an odd numbered row since these rows correspond to upper halves of panels.
- 5 needs to be in an odd numbered column since these columns correspond to left sides of panels.

It was an otherwise boring example, but you can check these properties against the first table we looked at.

What we need then is a formula or rule that helps us determine which rows and columns 5 will end up in based on the number of columns in the sheet. **WARNING: **This part will get a little complicated and mathy. Feel free to skip to the section titled “DARN IT!” if the math gets too distasteful for you.

Let’s assume our sheet is *N* cards across and has *M* rows of cards. Then somewhat generally our sheet looks like this. Almost.

I say almost because we haven’t yet accounted for our numbering restarting every time we reach 36. For example, when *N* = 18 and *M* = 10, we don’t want to see our sheet numbered 1-180. Exactly as in our first example, we want to see our sheet numbered 1-36 five times in a row.

A mathematical trick that will get us what we need is called modulus (or sometimes “clock arithmetic”). In this case, we just need to replace each sheet entry with its equivalent, modulo 36. (Technically, this still isn’t quite right since it would replace 36 with 0, but we can simply keep that oddity in our back pocket for now.)

Therefore, our new generalized sheet looks like this:

Now that doesn’t look awful, does it? Okay fine, don’t answer that! And of course, I’m about to make things even a little more complicated since I can’t do a heckuva lot with all those dot-dot-dots taking up most of the sheet. I’d rather have variables, even if it means going from two letters (*M* and *N*) to four.

For example, if I am in row *j *and column *k*, I want to be able to say the number in that spot is [(*j *– 1)**N *+ *k*] mod 36. That way, if I want to know if the card is of Dr. K, I just need to check to see if [(*j *– 1)**N *+ *k*] mod 36 = 5! Simple, right?

Now let’s go back to the two conditions necessary for Dr. K to occupy an upper left panel position: odd row and odd column, which algebraically just means *j *and *k *are odd.

Of course (!) when *j *and *k *are odd, then [(*j *– 1)**N *+ *k*] is also odd regardless of the value on *N*, so there is at least hope we will have solutions for correctly chosen values of *j* and *k* no matter the width of the sheet.

What about if we want Doc to appear in the upper right, which I’ll note we have not yet seen in any of our examples? The conditions for this are–

- 5 needs to be in an odd numbered row since these rows correspond to upper halves of panels.
- 5 needs to be in an even numbered column since these columns correspond to right sides of panels.

Algebraically then, we know *j *is odd but *k *is now even. Going back to our complicated looking expression, that makes [(*j *– 1)**N *+ *k*] even no matter what value of *N *we choose, and this guarantees we will never have [(*j *– 1)**N *+ *k*] mod 36 = 5.

In case your eyes are glazing over, this means Doc will never end up in the desired position, which in turn means we will never get our four variations no matter what size sheet we go with. We could still look at the two lower quadrants but after suffering such a fatal setback one might justly wonder what the point would be.

#### Darn it!

Well imagine that! No matter what size we make our sheet, there’s no way to arrive at four Doc variations following a methodical cutting strategy. Two is as good as it gets. And yet, I started this article by showing you four actual examples. So what gives?

There are really only two options, assuming we’re still in the “panels produced from uncut sheets” camp.

- The cutters didn’t follow a methodical approach, instead just cutting up sheets willy-nilly to achieve desired results.
- The sheets didn’t follow the same rules we did.

The first option is hard for me to swallow, but the second one is equally baffling. Yes, one could imagine laying out the sheets so that we didn’t simply have the complete set repeated over and over. For example, we might imagine the insertion of a double-prints within the set.

The problem is that even the insertion of a double print doesn’t generate new solutions unless it were only sometimes inserted. (For example, imagine the first instance of the set on the sheet included two card 10s, but none of the other instances did. Possible but weird, right?)

#### Cheating, PArt ONE

Okay, what the heck. Let’s just look at an uncut sheet, shall we? Perfect, let me just get out my microscope!

Kidding aside, the sheet has everything we needed to know. First off, it’s 20 cards across and 22 rows down for a total of 440 cards, which is enough to hold 12 complete sets with 8 cards left over. We can also (barely) see that the sheet doesn’t even come close to following the logical layout originally assumed and later dismissed. I’ll highlight this by placing a thick red border around every Nolan Ryan card since his card is both easy to spot and card #1 in the set.

Sure enough, there are 12 Ryans, but their appearances on the sheet become more and more irregular as we progress down the sheet. Life gets more sane once we add nine more rectangles.

Each of these 6 x 6 blocks is itself a complete set, always following a consistent sequence. For what it’s worth, that sequence is:

32-21-08-30-33-04

10-07-25-19-22-02

05-34-26-28-29-13

17-06-16-36-27-12

23-35-09-18-20-24

01-14-31-03-11-15

We can further simplify the sheet by making three more rectangles. The upper orange rectangle houses the first two columns of a set, the middle orange rectangle houses the middle two columns, and the bottom orange rectangle houses the last two columns. Collectively they add up to the tenth complete set in the diagram.

All that remains then is to understand the last four rows of the sheet.

Each of the green blocks at the bottom represents the upper four rows of a complete set, which the two yellow rectangles then complete perfectly. Finally, the pink rectangle gives us our eight leftover cards, which match up with what we might call the upper left corner of the set.

That’s all well and good, but of course what we really want to see is whether this sheet layout produces all four Dr. K variations. At least that’s what I’m here for!

One thing we can say without a doubt is that the actual sheet layout is nothing like our original assumptions. With all the cockamamie patchwork we might justifiably exclaim, “Aha! THAT’s the crazy kind of layout you need to get all those variations!”

But guess what?

Cutting the sheet methodically into 2 x 2 panels, we find only all twelve Dr. K panels yield only a single variation (see yellow stars), the exact one I bought.

Go figure!

#### Cheating, part two

The only thing for me to do was ask the seller, who specializes in oddball cards and has always been a great guy to do business with. While his answer may not hold for all 4-in-1 panels like mine, he let me know, sure enough, that the particular panels in his own inventory came from uncut sheets.

I then asked if he did the cutting personally since I’m still surprised someone would resort to the unorthodox method it would require to produce all variations. He let me know he was not the cutter and had received his rather extensive lot as part of a larger purchase he made.

Perhaps one of my two readers who’s made it this far has direct knowledge in this area, in which case I hope they share that knowledge in the comments. In the meantime, I’m left believing that somebody somewhere wanted variations so badly that they were willing to live with a little waste.

Either that or…

#### what really happened

So I was thinking about the easiest way to produce all variations for all players with a minimum of waste. Taking Doc as an example, we know he only appears in the upper right quadrant when methodical cutting is applied. However, moving him from top to bottom just requires skipping a row. Similarly, moving him from right to left just requires skipping a column.

Treating the blacked out portion of the sheet below as throwaway, a methodical cutting of the remaining sheet portions will generate a complete master set of all 36 x 4 = 144 panels. I’ve highlighted one of each Gooden panel to illustrated this. (Note also that my black lines could have been just about anywhere and still done the trick. They only requirement is that each quadrant remain large enough to generate a panel of the desired player.)

So yes, I do believe that’s what happened, and the only question that remains for me is whether the cutter planned all this or just stumbled upon it by accident. As for that I’ll offer that many uncut sheets end up with a bad row or two due to creasing. I’ll also offer that bad cuts here and there are possible when chopping up an entire sheet. What this suggests to me is there are at least two ways the original cutter may have unwittingly generated a master set all my math told me was impossible!