Magic: Exact Rarities

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Re: Magic: Exact Rarities

Post by Muldoon » Thu Mar 04, 2004 1:27 am

Stephen did point out that his estimations of starters and boosters are purely subjecture.  I am still curious if anyone knows the composition on an Alpha Starter and to refute if an Alpha booster pack did not contain 1 rare, 3 uncommons, and 11 commons.  Beta numbers would also be nice.  ;D

Once the numbers are determined, a pretty accurate booster and starter count can be constructed for both Alpha and Beta.  Since the oldest pack I have ever personally opened has been Legends (and only 1 of those), a little help is needed here.
Magic: The Gathering Limited Edition:
-------------------------------------

Sold in "Starter Decks" of 60 cards and 1 rulebook for $7.95 and
    in "Booster Packs" of 15 cards for $2.45

Decks contain:         (754.72 total cards per $100)
  2 Rare cards        ( 25.16 rares per $100)
 13 Uncommon cards    (163.52 uncommon per $100)
 45 Common cards      (566.04 commons per $100)

Boosters contain:      (612.24 total cards per $100)
  1 Rare card         ( 40.82 rares per $100)
  3 Uncommon cards    (122.46 uncommons per $100)
 11 Common cards      (448.98 commons per $100)

Here's the quantity info (thanks to motorcitymagic for that link).
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Re: Magic: Exact Rarities

Post by mintcollector » Thu Mar 04, 2004 1:34 am

Magic: The Gathering Limited Edition:
-------------------------------------

Sold in "Starter Decks" of 60 cards and 1 rulebook for $7.95 and
    in "Booster Packs" of 15 cards for $2.45

Decks contain:         (754.72 total cards per $100)
  2 Rare cards        ( 25.16 rares per $100)
 13 Uncommon cards    (163.52 uncommon per $100)
 45 Common cards      (566.04 commons per $100)

Boosters contain:      (612.24 total cards per $100)
  1 Rare card         ( 40.82 rares per $100)
  3 Uncommon cards    (122.46 uncommons per $100)
 11 Common cards      (448.98 commons per $100)

Here's the quantity info (thanks to motorcitymagic for that link).
Thanks for pointing this out.  I forgot to look at motorcitymagic's URL.  Now to start working some numbers....

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Re: Magic: Exact Rarities

Post by Archivist » Thu Mar 04, 2004 1:52 am

I ordered a Beta Starter and got an Alpha Rare. What happened?

When Wizards of the Coast, the makers of Magic, switched production from Alpha to Beta, they had some Alpha rares left over. These rares were mixed into the Beta print run and inserted into some of the Beta Starters. Because they did this, it is possible for you to purchase a Beta Starter and occasionally get an Alpha rare.
I found this quote while searching for something else. Anyone ever heard of this before?
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Re: Magic: Exact Rarities

Post by AXIOS » Thu Mar 04, 2004 2:38 am

I found this quote while searching for something else. Anyone ever heard of this before?
i did, if i remember well, it was in a very early isse of duelist or inquest
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Re: Magic: Exact Rarities

Post by Muldoon » Thu Mar 04, 2004 3:41 am

I think that quote was from the starcitygames faq , too lazy to check though. It's interesting though because it could actually make up for the "missing" 100 beta rares...
If they had as many as 100 spare sheets I'm pretty sure they wouldnt just discard them...
We need an early wizards or carta mundi employee to be our insider and verify this kind of info/qualified guesses ;D
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Re: Magic: Exact Rarities

Post by fvzappa » Thu Mar 04, 2004 3:49 am

Alpha rares were left over (why I don't know) & the original batch of Beta had some Alpha in it. I'm guessing at the time they could have cared less that the corners were different.

From what I understand, it was only rares, & only in starters.

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Re: Magic: Exact Rarities

Post by hammr7 » Thu Mar 04, 2004 8:12 am

Don't get too concerned over the discrepency between the 1100 copies of each Alpha Rare vs. the 3200 copies of each Beta Rare.  

If you check the Crystal Keep document, the author talks about rounding the numbers.  In fact the document states that the numbers were rounded to the nearest 500.  With the exception of Alpha and Beta Rares, every other number is rounded to multiples of 500.  

Apparantly the Alpha and Beta numbers were so low that they were rounded to the nearest 100 instead of 500.  This could explain why there were three times as many Beta cards produced, with a result of 1,100 Alpha copies and 3,200 Beta copies of each rare.  If the actual number of Alpha sets was at least 1,050, and no more than 1,083, then the Alpha number would round to 1,100.  If three times as many Beta cards were produced, then there would be between 3,180 and 3,249 copies of each rare, which would round to 3,200.
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Re: Magic: Exact Rarities

Post by mintcollector » Fri Mar 05, 2004 3:10 am

Based on CrystalKeep's numbers I found a point of interest.  We know that there are 116 R1's printed on a 121 card sheet leaving 5 land rares.  Crystal Keep reports 1100 R1's of each type existing, implying that 1100 Alpha R1 sheets were printed.  This means there are 133100 "rares" (an actual 127600 rares and 5500 lands in rare slots).  95 U1's on a 121 card sheet leaves 26 land uncommons.  CK reports there are 4500 U1's of each type, or 544500 "uncommons" (427500 actual uncommons and 117000 lands in uncommon slots).  74  C1's on a 121 card sheet leaves 47 land commons.  CK reports there are 16000 C1's of each type, or  1936000 "commons" (118400 actual commons and 752000 lands in common slots).  Adding up all the lands together (5500+117000+752000) gives us 874500 lands total.  CK reports 85500 lands of each type, so 2 land artworks of 5 basic lands types, gives us 855000 lands.  This means there is a 19500 land count discrepancy to account for.  Since there are only 5 land rares on a rare sheet, the difference here is negligible, so let's look at uncommon and common print sheet counts.  Since CK did rounding to the nearest 500, the worst case actual count for uncommon sheets can be 4251, which leaves 249 sheets at best.  There are 26 uncommon lands, so this could account for 6474 of the 19500 land discrepancy, leaving 13026 lands still unaccounted for.  If we assume that these are moctly commons, once again having a max of 249 common sheets due to the 500 rounding, we get 11703 lands for this.  This leaves 1326 lands still unaccounted for.  At 5 lands on a rare sheet there is no way we can account for the missing lands to be from rounding of rare sheets.  We can assume that up to 249 rounding can have occured from the 85500 lands being reported per type, or a 2490 count overall.  Here is the probable missing 1326 lands unaccounted for.  Since the numbers are pretty close, we can assume that there was 200+ rounding on uncommon and common sheets, pretty close to the 249 though, so I went with 240 off on each count.  So more realistic print run numbers are 1100 rares of each type, 4260 uncommons of each type, 15760 commons of each type.  Re-tallying the land up using these new numbers (1100 x 5, 4260 x 26, 15760 x 47) we get 856980 lands, which is still 1480 lands over what is reported, but a far cry more accurate  Refiguring the overall print run numbers (1100 x 121, 4300 x 121, 15800 x 121) we get 2565200 cards total, which still is close to the reported 2.6mil cards reported overall.  The numbers make sense and the math seems solid so I think these are closer to what is actually out there for Alpha.

Now to the booster to starter ratio question.  It is often difficult to reconstruct algebraic equations based on severely  rounded data, so I took a different route to showing some stats.  We know that there are 2 rares, 13 uncommons, and 45 commons in a Alpha starter.  There are 1 rare, 3 uncommons, and 11 commons in an Alpha booster.  We also know estimated print runs of each type..    The base to start from is the rare count.  Let's assume that all 133100 rares are in boosters.  This means there would be 399300 uncommons (3 x 133100) and 1464100 commons (11 x 133100) in those boosters.  Now lets look at what the numbers would be if all rares were in starters.  There are 2 rares in a starter, so this would mean 66550 starters.  There would be 865150 uncommons (13 x 66550) and 2994750 commons (45 x 66550) in those starters.  using CK's numbers and not my new ones calculated in the previous example, we get 554500 uncommons (4500 x 121) and 1936000 commons (16000 x 121).  So:

All boosters===CK's reported numbers===All starters
R: 133100===133100===133100
U: 399300===544500===865150
C: 1464100===1936000===2994750

So if you think of the above example as kind of a slide bar where CK's numbers fall somewhere in between both extremes, you will see that the numbers lean towards more boosters than starters.  Let's now figure how far these numbers deviate into each range.  CK's uncommon number of 544500, is 36.36% above the All booster uncommon number (544500/399300).  CK's common number of 1936000, is 32.23% above the all booster common number.  This means that the real number lies in between 32.23% and 36.36%.  Let's assume 33.33% for simplicity's sake.  Since we see that the numbers show more boosters than starters and have assumed an easy to calculate 33.33% deviation off of all booster, we can see a ratio of 66.67% boosters to 33.33% starters, or in easy to understand layman's terms, 2 boosters for every starter.    With 2 rares in each starter and 1 in each booster, this means half the rares went to boosters and half to starters, with there being an estimated 66550 Alpha boosters and 33225 Alpha starters.  This proves Crystal Keep's estimates of 50/50 for boosters to starters is incorrect simply by using the site's own data.  I'll need to work on Beta later, but I found these 2 points I made pretty interesting and wanted to share.
 

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Re: Magic: Exact Rarities

Post by fvzappa » Tue Mar 09, 2004 5:02 am

Whew, mintcollector, heavy math for even this math scholor.

Don't cramp your brain doing Beta. But thanks for the info.

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Re: Magic: Exact Rarities

Post by mintcollector » Tue Mar 09, 2004 7:42 pm

I'm still working on it in my spare time which has been next to none the last few days.  I have even discovered some other interesting items as well I'd like to post on.

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Re: Magic: Exact Rarities

Post by Erl00 » Tue Mar 09, 2004 8:40 pm

Though I don't post, I'm following this thread with attention.

I like maths and I've always been interested in making links between rarity factors / number of cards issues, ratio booster starters, etc.

I just have no spare time at the moment to make estimates / to confirm or inform what's been posted by others.

:(
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Re: Magic: Exact Rarities

Post by hammr7 » Tue Mar 09, 2004 10:24 pm

mintcollector,

I am also a math junkie.  When you have three different "variables" (rares, uncommons, and commons) you can never guarantee that you will have a perfect fit of ratios between the starter and booster packages.  Your analysis above starts with the premise that all rares go into packages, and builds proportions from there.  There are alternate initial assumptions that might provide a little more accuracy in determining the original mix.

A problem with any analysis, as you have noted, is the rounding errors.  Another way to look at the Alpha numbers, given this rounding concerns, is to start with the biggest numbers, which are hopefully less affected.  The number of commons, even with the bigger "500" unit rounding, will be closer to perfect than will the rares with their "100" unit rounding.  The uncommons will be almost as accurate (statistically) as the rares.

There appears to be justification for working with the uncommon-to-common ratio as a starting point, rather than the assumption of all Alpha rares being utilized in Alpha product. It has been stated above that Wizards had excess Alpha rares which found there way into Beta product.  If a "perfect" consumption of commons and uncommons results in an excess of rares (within the allowable variances of +/- 250 cards for both the common and uncommon groups), then the resulting product mix is supported by anecdotal evidence.

Granted, any of this is splitting hairs given the number of uncertainties involved.  We are assuming production of perfect sheets when in all likelyhood there were individual cards scrapped from each rarity.  We are assuming that Wizards utilized all produced cards for retail product, when in all likelyhood some was pulled for various "corporate" functions (marketing, sales, QC, etc.)

Like you, I have been time-constrained lately.  I'll attempt some sort of analysis when time allows (hopefully next week)
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Re: Magic: Exact Rarities

Post by mintcollector » Wed Mar 10, 2004 12:18 am

hammr7:  I am fortunate enough to know people who were at the WOTC investment/corporate product kick offs.  Carti Mundi poker decks were given out to investors as well as other promotional details (I know because he gave me one of his sealed decks for free), but individual cards were never handed out as promotional items to the best of their knowledge.  We cannot rule out the quality control or even internal sampling may be the end result of some product.  We can only work with what we know.  We should assume that 99.99% of product was used for sales, thus making calcaultions for 100% should not be far off in calculations.

I am also aware the Alpha rares made it into Beta product, but I doubt this was very widespread.  Enough to where I think factoring it in makes for unecessary complexity that could simply be ignored.

I ask anyone following this thread that instead of conducting different work, try to refute what I have surmised as I think my findings are fairly accurate.

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Re: Magic: Exact Rarities

Post by hammr7 » Thu Mar 11, 2004 11:58 am

I have had the opportunity to review your numbers, and have come to the following conclusions.

1.  I think that your analysis of Alpha basic lands is  correct in determining the larger number of lands, but it is not necessary to modify the original numbers of Alpha sheets to explain their existence.  If you read the Crystal Keep page describing Alpha and Beta sets, you'll note that they aren't separated, but are instead combined on a "Limited" page.  While the page describes, in detail, the differences between the Alpha and Beta printings, it only lists the Beta numbers for calculating the percentages of lands in the rare, uncommon, and common slots.  

If you apply the Beta percentages to the listed number of rare, uncommon, and common Alpha sheets, you arrive at a value of 85,740 copies of each basic Alpha land.  When rounded to the nearest 500 (as stated in the original memo), this value becomes 85,500, which is the original value in the Crystal Keep memo.  

I therefore propose that whoever calculated the number of basic lands in this original Crystal Keep memo used the Beta percentages by accident in determining the number of Alpha basic lands.  It means that you correctly calculated the actual number of Alpha basic lands.  

This conclusion allows us to maintain all the other listed Alpha information from the memo.  It's a simple way of solving your "excess" land problem.  And this revised number of basic lands doesn't impact any other calculations; since basic lands are simply the surplus of cards that aren't rare, uncommon, or non-land commons.
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Re: Magic: Exact Rarities

Post by hammr7 » Thu Mar 11, 2004 12:00 pm

2.  In determining the number of cards that go into boosters vs. starters, there are a few basics definitions that need to be specified.  In calculating percentages for each type of package, do we consider total cards as the basis for percentage, or do we consider total number of rares in each type of packaging, or do we consider total units of boosters vs. total units of starters?  In your calculations you assume the latter, but I haven’t yet determined whether Crystal Keep’s memo writer also chose that basis.  It may be that your calculations support Crystal Keep’s conclusions, but with the two of you speaking different languages.

I am concerned that your attempt to determine these percentages was based upon complete utilization of rares, despite anecdotal evidence to the contrary.  While I don’t necessarily put credence in such evidence, I am loath to contradict it.   Furthermore, I think that the Crystal Keep Alpha and Beta data gives us an indication that either Alpha rares were in excess, or there were less than 1100 original rare sheets.  If you review the ratios between the Alpha and Beta data, the uncommon and common Beta sheets are exactly triple the Alpha sheets.  However, for rares, the Beta number is 100 sheets less than triple the Alpha number.  

This can logically be explained by either of two scenarios.  The first is that there were excess Alpha rares, so less Beta rares were needed.  The second is that the Beta “rareâ€￾ run had exactly triple the sheets of the Alpha run, and the apparent numerical difference is a result of rounding.  The second case can only be true if the actual number of Alpha sheets is between 1050 and 1083. In this case, triple the number ranges from 3150 to 3249, which is rounded to 3200.  Coincidentally, the first case is true for approximately the same range of actual sheet utilization (1100 Alpha rare sheets made, but only 1063 to 1087 used).  Remember that the Beta run was not a different series of cards, but an improved continuation of the Alpha printing (to meet overwhelming consumer demand).  Wizards could have anticipated their leftovers of each Alpha rarity when the placed their supplemental orders for Beta.  The fact that Crystal Keep predicts maximums of 1100 Alpha sets and 3200 Beta sets (when rares are the limiting factor) makes me believe those were the numbers of sheets, and therefore there was an excess of Alpha rares to be matched with some of the Beta commons and uncommons.

Where you chose complete rare utilization for Alpha, there are many other possible models to evaluate.  These include:

a)      Minimizing the total of all remaining Alpha cards, regardless of rarity.
b)      Minimizing the total leftover commons and uncommons, so long as some rares remain.
c)      Maximize the leftover rares, less the sum of the other leftovers.

Each of these scenarios can be evaluated with an expectation of 1100 rare sheets.  The first two scenarios (plus yours) can also be evaluated with expectations of 1087, 1070, and 1050 rare sheets.  Optimal solutions to these scenarios will provide a number of feasible solutions (some of which should be boundary solutions) to the actual distribution of boosters and starters.

There is one more constraint that I believe is necessary.  Your calculations implicitly predict optimality down to fractions of a card.  In the real world, cards must be integer (whole).  Furthermore, cards must be grouped into packs, packs into boxes, and perhaps boxes into cases.  Therefore any solution must anticipate full boxes at the least, and full cases at the most. These constraints can be incorporated into a linear programming model, which can be implemented through the Solver add-in on Excel.  According to Crystal Keep, Alpha had 36 boosters per box,  and 10 starters per box.  I don’t know how many boxes of each were in cases.  If you can supply me with those numbers, I will run the different scenarios and provide the calculated results.  The results will be provided in boxes (or cases) of boosters and starters, and leftover amounts of commons, uncommons, and rares.
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