You should be able to find this piece by looking up any of the collectible minifigures on Bricklink and looking at the set inventory.
This piece is the Tile, Modified 3 x 4 with 4 Studs in Center (88646).
It is classified on Bricklink under the category 'Tile, Modified'.
I know the question was phrased as a sort of math problem, but most LEGO fans solve it as a sort of LEGO problem. The bags are opaque, so you cannot see what's in the bags, but almost everywhere they are sold you can feel the bags, and you can determine what figure is in the bag. With the exception of the recent Soccer Team figures, collectible ...
If the distribution is assumed to be completely random (not necessarily true), and of a large enough group to make your purchases not effect the total probability (likely) then this is an alternate of a classic math problem known as the Coupon Collector's Problem:
There are many ...
I was really stumped on this when I posted the question because there didn't seem to be any place to attach to. But then I realized the neck could be a starting place. I tried some things with hinges and plates, but decided to simplify and I ended up with:
With only 4 pieces I was able to get this to work out pretty nicely. From neck to plate there are:
For any CMF's with skates, I put the figure on the stand with no footwear, and attach the footwear to the holes behind their legs. That way their accessories are there, and stuck to the figure, but the figure is on its stand!
It's known as the "Coupon collector's problem". To get all 12 you would on average need to buy 38. Buying a whole box (or finding someone who did and buying a set from them) seems like a much more attractive solution.
Some of the minifig parts that are officially made in China (often including the Collectable Minifigures) don't have the copyright/logo stamp on the inside toe - I noted that in a comment to my answer on What does the imprinted information inside bricks mean?
Every box contains the same distribtion of minifigs. For example see this review on www.eurobricks.com of series 13 has the same distribution as my box. (links to his other CMF reviews are on the bottom of the first post)
For most series boxes contains 3 full sets and 12 remaining ones.
Series 9, 10 & 11 contain only 2 complete sets, with 2, 4 or 6 ...
If you are on good terms with your lego store, you could buy an unopened box with 60 bags which will guarantee you to have 5 of each. You shouldn't have much difficulty selling the other series, even if the bags are opened, to get your money back.
Spring-boarding off your nice work-around, I would suggest a further aesthetic tweak with the symmetrical re-positioning of the support shaft, and as a bonus, eliminating the 88646 tile all together, if it's not being used.
I don't have penguin accessories in my parts library, but here are some pics to help visualize my techno babble;)
While looking up ...
On thebrickblogger.com (February 9, 2017) there was a call to report issues with minifigures. Reading the answers I get the impression that issues with clutch-power, cracks and other problems are occurring with all sorts of minifigures, not only with the collectible series. This would mean that it is not a persistent problem with Chinese factories:
It depends on licensing as well. Note that they have made sets for Ultimate Spiderman. The X-Men designs all follow their comic counterparts. Loki, Thor etc follow their Movie style counterparts.
This suggests that they have the license for the comic and animated styles, but Fox and Sony retain the rights for the movie looks of the characters.
I don't think that such a set exist. The most complete database are the ones indicated by Guilherme Branco Stracini, BrickLink and BrickSet, but both of them tell in their footer that the content is copyrighter so I am not sure that, especially if you intend to sell your app, any of them will allow you to use their content.
On the other hand, BrickLink ...