General Pack Size

I'm sure there's a thread for this already so if some mod wants to merge this discussion in with that, be my guest.


Some of us were having a discussion on the Brainstorming discord today about pack size for a traditional 8 player draft, some folks like the traditional 15 cards x 3 per player, I'm a special snowflake and run 4 packs of 16, I know some folks have tried the 20 card packs thing that wotc broke out in commander legends; blah blah blah lots of options.

Which method do you like most? Have you tried others that you found worked well? Have any been super shitty? Why?

@landofMordor even asked the question today "why do packs need to wheel at all?" and it's gotten me thinking about trying a switch to 8 packs of 8! Interested to hear what you nerds think.
 

landofMordor

Administrator
Love it. I was talking in my cube's thread about some considerations for smaller packs, namely that the human brain takes longer and is less satisfied with more than ~5 choices. This isn't an issue late in draft, or if you have strong predispositions towards a certain color, because that filters out a lot of possible options in a 15-card pack. But it can be a big deal if you're trying to first pick between 16 roughly interchangeable, high-quality effects, or if you're weighing the relative merits of cards like Vindicate vs. Growth Spiral vs. Wild Nacatl that all do very different things equally well.

And that got me thinking that I don't even know what is gained with packs wheeling -- sure, it makes signaling a bit clearer, but I tend to figure out my signals on picks 4-7, not 8-15.

The only concrete argument I can think of for bigger packs is that it's a larger sample size, which will tend to decrease variance in pack collation, color distribution, etc. I'd love to hear other opinions or ideas, though. I even hear that some of our number are trained statisticians and can fill out my layman's interpretation of sampling size (;
 

Jason Waddell

Administrator
Staff member
I'm sure there's a thread for this already so if some mod wants to merge this discussion in with that, be my guest.


Some of us were having a discussion on the Brainstorming discord today about pack size for a traditional 8 player draft, some folks like the traditional 15 cards x 3 per player, I'm a special snowflake and run 4 packs of 16, I know some folks have tried the 20 card packs thing that wotc broke out in commander legends; blah blah blah lots of options.

Which method do you like most? Have you tried others that you found worked well? Have any been super shitty? Why?

@landofMordor even asked the question today "why do packs need to wheel at all?" and it's gotten me thinking about trying a switch to 8 packs of 8! Interested to hear what you nerds think.

Anecdotal, but we have in the last months been testing various asynchronous draft formats, so that we can do a decently proper forum draft that doesn't take forever.

My first take was 'Triangle Drafting' (each player starts with a triangle of 45 cards: one row of 9, one row of 8, ...., one row of 1). The format is that you take one card from each row and pass the rest on. After one test this was switch to Rectangle Drafting (five rows of 9). We also implemented a special randomization method for the backs so that signalling worked much, much better.

One of the driving factors was the smaller average pack size in Triangle Drafting made for some awful drafts. Generally speaking, the more picks you make from packs that have 1, 2 or 3 cards left in them, the sloppier your decks will be.

My personal opinion is that wheeling twice is not really that important. I've never cared that I didn't get 2 cards from one of the 8 packs. I would think that 8 packs of 8 would result in weaker decks with more 'random' cards ending up in the player's pile of drafted cards.
 
If packs don't wheel, you have less of a traditional draft experience and more like a "advanced sealed" style of play. There is a lot less of the traditional interplay that wheeling provides. This isn't a condemnation of non-wheeling strategies, it just feels different to do.

My most common form of drafting is Tenchester (with 4 players). 10 packs of 10, ditch 6 from each pack. Basically each player gets a sealed pool of 250340 cards to sort through to find 40 candidates, and the entire cube is shown each draft.
 
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landofMordor

Administrator
Also, to answer the original question: I've tried 5 packs x 9 cards, or 4 packs of 10 cards, but only when drafting with less than a full 8-person pod. I made this change because I definitely didn't like it when 15 cards wheeled 3 times around the table of 4 drafters. I found it boring, and subjectively/qualitatively noticed that narrower cards tended not to find a home, and might as well have been removed from the draft pool to save everybody time. This did fix an upper limit on *too many* wheeled picks.

So maybe my personal upper limit for an 8-person draft would be 24 cards in a pack. But, just to be safe, I'd probably not go above 16.
 

landofMordor

Administrator
If packs don't wheel, you have less of a traditional draft experience and more like a "advanced sealed" style of play. There is a lot less of the traditional interplay that wheeling provides. This isn't a condemnation of non-wheeling strategies, it just feels different to do.

My most common form of drafting is Tenchester (with 4 players). 10 packs of 10, ditch 6 from each pack. Basically each player gets a sealed pool of 250 cards to sort through to find 40 candidates, and the entire cube is shown each draft.
Your first paragraph isn't intuitive to me. What is the interplay of wheeling? Or, how does not getting a wheel lead to Sealed-like play? (genuinely curious :))

When I draft, picks 1-8 are me reading draft signals to try and find a lane, picking colors more or less at random. Then, picks 9-15 are me carving out my lane based on what everybody else didn't pick at quasi-random. It kinda feels like the same thing as an entirely new pack. (Maybe I'm revealing my lack of Draft skill, lol!)
 
Your first paragraph isn't intuitive to me. What is the interplay of wheeling? Or, how does not getting a wheel lead to Sealed-like play? (genuinely curious :))

When I draft, picks 1-8 are me reading draft signals to try and find a lane, picking colors more or less at random. Then, picks 9-15 are me carving out my lane based on what everybody else didn't pick at quasi-random. It kinda feels like the same thing as an entirely new pack. (Maybe I'm revealing my lack of Draft skill, lol!)
You don't get the later-pick signaling from other players snatching things you wanted to wheel, and the inverse is also true, you don't get that gauge of openness from a risky wheel that does in fact make it back to you. You don't compete on card selections the same way, because everyone only sees cards once. Additionally, you get 8-10 completely randomized first picks as opposed to 3-4.

This divide in draft experience is even bigger when dealing with fewer than 8 players, because traditional packs wheel around more than 2 times in that case. (5 or fewer people is 3+ complete cycles)

Obviously there will be some upstream signaling (if someone is scarfing red spells in the seat ahead of you or something). The sealed analogy will never be perfect because, well, it isn't literally sealed. Basically, it's a more individualized experience and the decision of others are felt less at your own seat than "normally".

This is doubly the case with draft styles like Tenchester where you bloat the packs and ditch extra cards. You don't get as good a signal off the 1 to 3 previous picks as you would with something like packs of 8 at an 8 person table, where you can still (somewhat) see openness-signals after a few picks.
A tenchester drafter gets to see 100 cards totally randomly selected without anyone else's input bias, and make 10 first picks off that. That's already more cards than an entire sealed pool even is.

I think you mean 40 packs of 10.
yeah 10 packs of 10 per person so 40 total. I thought that fit the theme of how the packs were being described ("3 packs of 15"; people weren't saying "24 packs of 15")
 
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Jason Waddell

Administrator
Staff member
yeah 10 packs of 10 per person so 40 total. I thought that fit the theme of how the packs were being described ("3 packs of 15"; people weren't saying "24 packs of 15")
Oh, I see what you mean. I wasn't thinking of Tenchester as an exclusively 4-person format, so if you would run it with 3 people you'd still want like 40 packs, which changes the packs per person math.
 

Onderzeeboot

Ecstatic Orb
@landofMordor even asked the question today "why do packs need to wheel at all?" and it's gotten me thinking about trying a switch to 8 packs of 8! Interested to hear what you nerds think.
I do have thoughts about that! :)

Love it. I was talking in my cube's thread about some considerations for smaller packs, namely that the human brain takes longer and is less satisfied with more than ~5 choices.7
I feel you can't extrapolate this into meaning we should drastically reduce our booster size, since all those cards actually serve a function. Choice is, in fact, the most important function of offering a larger booster size. A smaller booster, say one with 8 cards, will offer much less options to pick a direction. A spike-y cube like Funch's might suffer less from this, since, I believe, his pieces are more interchangeable, his drafters more archetype-agnostic, and the way his cube is set up more amenable to picking objectively flexible and high value cards and/or mana fixing. A more traditional Riptidian cube with narrower archetypes will probably be harder to successfully navigate if the breadth of a pack is more narrow, and crucial pieces are less likely to make their way to you. I know I often find myself feeling out the right archetype to dive into and hoping to wheel specific pieces. That said, maybe 15 cards is too much. It's just what we grew up with, so who knows if there isn't a better alternative. My thoughts on the matter are all very much speculative and rooted in what I know.

One thing I do find aesthetically pleasing, is that 15 is 2n-1 if n = 8, i.e. with a full pod of players. For this reason, I have drafted 4 packs of 11 cards at least twice when I could only muster 6 drafters, and those drafts worked out eerily close to what you would expect from an 8 man 3 x 15 card pack draft.

Quickly brainstorming off the wall ideas for a no wheel environment, I do have a quirky idea for a different pack size / draft format.
  • Pack size n+3, where n is the number of players (so 11 for 8 players, 7 for 4 players).
  • First player gets two picks.
  • Players 2 through n-1 get one pick, as normal.
  • Last player picks one of three card and burns the other two face down. (These cards are removed from the draft.)
If you want you can introduce open information to assist signalling. Either the first player reveals one of their two cards, or the last pick is revealed (but not the burned cards, as that would almost assuredly cause information overload).
 
A more traditional Riptidian cube with narrower archetypes will probably be harder to successfully navigate if the breadth of a pack is more narrow, and crucial pieces are less likely to make their way to you.
This tidbit is one of the main reasons I like the Tenchester style. The packs are maxed out on size (in the constraints of the style), so you have decently broad packs to select from. And you get to see a huge fraction of the cube at each seat, which helps with navigating narrower archetypes. It all but guarantees niche playables will make it to the person that wants it, for instance. The entire cube is drafted and a niche card won't be picked out from under the desired drafter's hands (usually):

A tenchester drafter is handed 250340 unique cards, and 100% of the 400 cube is drafted.
A 4-player cube playing 3 packs of 15 sees only 162 unique cards and only 45% of the same 400 cube is drafted.

The downside is of course you can't wheel a second card you want out of a pack, so redundancies on some effects do get a little more important.

I'm really interested in your theoretical draft style, since the "two first picks" basically negates that Tenchester downside.... I'm considering maybe two last picks when dealing with a Tenchester pack style?? That would assuage some of the balance issues one might get with 20 first picks, while still giving each player a chance to grab that removal spell that otherwise would get tossed for the on-theme archetype enabler. And 50 card pools instead of 40; still in realistic bounds of a draft pool. hm............

EDIT: updated my unique card total. Somehow missed the "9" (10+9+8+7 x 10 is 340 unique cards viewed)
 
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I cubed with 3 packs of 16 from 2010-2013. Twice I tried going up to 4 packs of 16 (try 1 is here), and what happened was that decks became much stronger, and it became much harder to make cuts. A lot of the questionable playables and niche strategies simply became unplayable, because there was no incentive for trying to make use of those late picks. It was an interesting experiment, but we did not like it overall because it felt less "limited", less "build something that somehow works while being scrappy" and more "know what the best cards are and play them". It didn't reward players for being creative and squeezing the most out of their seat, but instead empowered them to follow a good-stuff line. I imagine that increasing the non-basic density would have offset this feeling, and at the time I had a 12.5% non-basic density.

I went back to the traditional 3 x 15 when playing with a playgroup that was already invested in Magic because it was closer to retail draft and more comfortable to them. As people are saying, 15 cards is a lot, and when the power band is flat it's a paralyzing decision because there aren't 2 or 3 power outliers that are "first pickable".

There's this dimension of tweaking how the cards are distributed in packs, but a more impactful dimension is increasing the pool size. I've given thought to going up by a booster and adding a lot of narrow cards, nonbasics, lessons, etc, but ultimately my draft sections already last 45 min - 1 hour, and going up a pack would not be a good idea.



On a tangent, a 2-player variant that I tested last year and was pretty happy with was based on breaking the pattern that choices are front loaded in regular drafts (more options early in the packs, fewer options later in packs and in the draft). For that, I used packs that get bigger in size as the draft goes by, since the more picks have been made, the more locked into colors and archetypes you are, so a bigger pack size counterbalances that and maintains the number of actual options more constant through the picks.

https://desolatelighthouse.wordpress.com/2020/12/21/pyramid-draft/
 
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Man... that drafting style is awesome! It's a little hard to grok at first, but I understand the underlying design rationale you have. That has me thinking about trying a size-shifted Tenchester style... something like:
1619813684958.png
(Per player, so 4x each row)
This still hits the 100/400 my cube would want, but borrows from your concept of minimizing decision overwhelm up front while keeping the decision density more linear across packs. Players are still getting the 340/400 unique cards in their pool, so nothing is lost there. Add in the "super picks" feature and bam.
 

landofMordor

Administrator
I feel you can't extrapolate this into meaning we should drastically reduce our booster size, since all those cards actually serve a function. Choice is, in fact, the most important function of offering a larger booster size. A smaller booster, say one with 8 cards, will offer much less options to pick a direction. A spike-y cube like Funch's might suffer less from this, since, I believe, his pieces are more interchangeable, his drafters more archetype-agnostic, and the way his cube is set up more amenable to picking objectively flexible and high value cards and/or mana fixing. A more traditional Riptidian cube with narrower archetypes will probably be harder to successfully navigate if the breadth of a pack is more narrow, and crucial pieces are less likely to make their way to you.
Agree that it's far from QED. I just think it's a valuable consideration, and goes to show that more choices =/= better choices.

In my cube, I have a (16-card) lands asfan of 3.6 and an 8-card asfan of half that, 1.8. So you're right in saying that an 8-card pack doesn't give me choices between 4 lands, only 2. The same would apply to asfans of cantrips, removal, creatures, etc. But I get twice as many first picks from those classes of cards...so it seems to me that the net effect is neutral (with the caveat that smaller sample size would lead to more variance). The same could be said for synergy pieces like Young PZ or Delver of Secrets -- the as-fan in a single pack is halved, but you get twice as many packs. (And I don't think it's true that the synergy pieces of a Riptidean cube are any less interchangeable than a cantrip in my cube -- the rates change, but not the as-fan of the critical mass support pieces.)

So I don't think it necessarily follows that smaller pack = harder to pick up deck's pieces. I haven't done the math on that, but probabilities like this are rarely intuitive. I'll see if I can dig something up...
 
Personally I'm a big fan of large pools, but that's a specific facet of the goals for my environment - testing and improving the skill of my players, and making drafting and deckbuilding a legitimate challenge and engaging part of the experience alongside constructed-esque decks.
 
So I don't think it necessarily follows that smaller pack = harder to pick up deck's pieces. I haven't done the math on that, but probabilities like this are rarely intuitive. I'll see if I can dig something up...
Smaller packs, if drafted per normal means (draft all cards until gone), reduce the number of unique cards that a player sees. This combines with higher overall pack power variance and less ability to wheel cards. Overall a drafter's ability to consistently seek out and chase niche archetypes is definitely reduced.
Some math examples below. First is standard 8 man draft. Second is Funch's originally stated 4x16 packs. Third is a 45 card pool shifted to 5x9 packs. Fourth is Funch's draft halved to 8x8 packs. The total unique cards seen is summed bottom right of each chunk. The 45 card pool has a 20% reduction in unique card sightings, while the Funchian shift is a 28% reduction as well as total loss of ability to wheel.
1619816880291.png

Here's a separate example with a 48 card pool, which is one of the most divisible numbers I could think of. The total unique sightings nose dive below a pack size of 12
1619817624627.png

EDIT: As has been pointed out I should have been a tad clearer on how I arrived at the unique card count.
Basically:
For one round of drafting it's (15 + 14 + 13 + 12.....) until you've gotten back to your starting pack, at which point you aren't seeing new cards. For a 15 card pack you see 92 cards in an eight drafter pod. With three draft rounds that's 92*3=276 total unique sightings.

Less basically:
Draft pack card counts are "triangular numbers" the size of the starting pack (T15 for a standard pack), that is to say they are sums of sequential integers. However, not every card in every pack is unique to a player the entire draft, because the packs they've already seen wheel back to them. These repeat packs make their own, smaller triangular number the size of the pack minus the number of drafters (T7 in a standard draft). To get the unique card count it's as easy as subtracting that second triangular from the first (T15-T7; this is not T8, as convenient as that would be). Below is the math I used in my excel sheet. One potentially unintuitive thing that you end up seeing is that the players collectively see a total number of unique cards greater than the number of cards in the cube. For an 8 man standard draft, the pod sees a whopping 2,208 unique cards, compared to just 360 cards in the cube. This might seem impossible at first glance, but because each player has a unique set of unique cards, that total is meaningless, so don't let it trip you up!
Code:
Where P = the pack starting size
Where D = the number of drafters
Where U = the unique card count

U = (P*(P+1))/2 - ((P-D)*(P-D+1))/2

If you want to be fancy and simplify the equation:

U = -0.5D^2 + P*D + 0.5*D
 
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Jason Waddell

Administrator
Staff member
I mean, taken to their extreme, 1 card packs are just random cards and infinitely large packs are Rotisserie drafting. All other pack sizes fall along this spectrum, and as you decrease pack size you increase randomness.
 

Onderzeeboot

Ecstatic Orb
Smaller packs, if drafted per normal means (draft all cards until gone), reduce the number of unique cards that a player sees. This combines with higher overall pack power variance and less ability to wheel cards. Overall a drafter's ability to consistently seek out and chase niche archetypes is definitely reduced.
Some math examples below. First is standard 8 man draft. Second is Funch's originally stated 4x16 packs. Third is a 45 card pool shifted to 5x9 packs. Fourth is Funch's draft halved to 8x8 packs. The total unique cards seen is summed bottom right of each chunk. The 45 card pool has a 20% reduction in unique card sightings, while the Funchian shift is a 28% reduction as well as total loss of ability to wheel.
View attachment 3900

Here's a separate example with a 48 card pool, which is one of the most divisible numbers I could think of. The total unique sightings nose dive below a pack size of 12
View attachment 3901
You don't really explain how you get to the number in the 'total unique' column, but I think I figured it out. Assuming 8 drafters, for a 15 card pack you are going to see 15 (your own booster) + 14 (the booster your neighbour passes to you) + 13 (etc.) + 12 + 11 + 10 + 9 + 8 (the booster of your other neighbour that went around the table before finally arriving in your hands) unique cards. The 9th booster is your own booster being passed back to you, so no further unique cards can be found in that (or further) packs in the round. In total this means you see 92 unique cards per booster you open, and since each player opens three boosters in the default 3x15 draft, you end up with 276 unique cards seen. For the 5x9 draft this sum amounts to 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 = 44 unique cards per pack opened, which amounts to 5 x 44 = 220 unique cards seen in the draft.

This is interesting math, because completely ignoring the aspect of wheeling cards, it clearly shows you simply get to see less unique cards per draft. Also interesting to note is that Funch's 8x8 idea actually works out to a number of unique cards seen that's very close to a traditional 3x15 draft :)
 
You don't really explain how you get to the number in the 'total unique' column, but I think I figured it out. Assuming 8 drafters, for a 15 card pack you are going to see 15 (your own booster) + 14 (the booster your neighbour passes to you) + 13 (etc.) + 12 + 11 + 10 + 9 + 8 (the booster of your other neighbour that went around the table before finally arriving in your hands) unique cards. The 9th booster is your own booster being passed back to you, so no further unique cards can be found in that (or further) packs in the round. In total this means you see 92 unique cards per booster you open, and since each player opens three boosters in the default 3x15 draft, you end up with 276 unique cards seen. For the 5x9 draft this sum amounts to 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 = 44 unique cards per pack opened, which amounts to 5 x 44 = 220 unique cards seen in the draft.

This is interesting math, because completely ignoring the aspect of wheeling cards, it clearly shows you simply get to see less unique cards per draft. Also interesting to note is that Funch's 8x8 idea actually works out to a number of unique cards seen that's very close to a traditional 3x15 draft :)
Yeah, probably should've explained that, but you are correct; its adding up pack card counts until you get your starting pack back, summed across the number of packs opened.

I didnt notice the similarity in cards seen between those two styles, that is interesting. Funchs cube is at least 512 however (8x8x8), so the percentage of total cube seen with 8x8 is 56%, while the standard 3x15 drafter sees 77% of a 360 cube.
 
Personally I'm a big fan of large pools, but that's a specific facet of the goals for my environment - testing and improving the skill of my players, and making drafting and deckbuilding a legitimate challenge and engaging part of the experience alongside constructed-esque decks.
I didn't read any nerd math yet, but I think this nearly sums up the concept. More cards means more choices. That's more skill testing and will improve card quality, but also more time consuming.

As a result, the "right" answer will depend on your playgroup. If your mates know every card and synergy without reading anything, go ahead and draft a ton of cards. Your decks will be more streamlined for it.

If you have newbies who take more time to analyze things, probably stick to 3x15. Maybe even give them a small pool of starting cards and have them draft some other number if it's posing a real problem. The latter doesn't seem likely to be an issue, but it does open up some design avenues as well as possibly saving some time in the draft portion.
Do you know where pyramid drafting is from?
Egypt???
 

landofMordor

Administrator
You don't really explain how you get to the number in the 'total unique' column, but I think I figured it out. Assuming 8 drafters, for a 15 card pack you are going to see 15 (your own booster) + 14 (the booster your neighbour passes to you) + 13 (etc.) + 12 + 11 + 10 + 9 + 8 (the booster of your other neighbour that went around the table before finally arriving in your hands) unique cards. The 9th booster is your own booster being passed back to you, so no further unique cards can be found in that (or further) packs in the round. In total this means you see 92 unique cards per booster you open, and since each player opens three boosters in the default 3x15 draft, you end up with 276 unique cards seen. For the 5x9 draft this sum amounts to 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 = 44 unique cards per pack opened, which amounts to 5 x 44 = 220 unique cards seen in the draft.

This is interesting math, because completely ignoring the aspect of wheeling cards, it clearly shows you simply get to see less unique cards per draft. Also interesting to note is that Funch's 8x8 idea actually works out to a number of unique cards seen that's very close to a traditional 3x15 draft :)
THANK YOU for explaining that math, lol. I was like "wait, 15*8=...", cue the meme of the lady calculating conic shapes.

@sigh thanks for doing that! Makes a lot of sense now that I see it. I didn't want to go on gut instinct, which is so often wrong in probabilistic calculations, but was busy at work and couldn't run the numbers myself. That's nice work, clearly illustrated.

Okay, I'm now totally on board that larger packs = more unique cards seen. We can assume more unique cards = easier to put together small synergies. However, it remains to be seen what the *optimum* cards seen is for each designer's goals. Is 400 unique cards too many? Is 288 too few? (And, the thing that might be harder to address by math -- what is the game design value/importance of wheeled cards relative to no wheels?)
 

Onderzeeboot

Ecstatic Orb
Adding a quick calculation for my brainstormed draft format.

With 8 players, each player sees 11 + 9 + 8 + 7 + 6 + 5 + 4 + 3 = 53 unique cards per booster opened. With 9 cards drafted per pack, you want to crack 5 packs for a 45 card pool, meaning each drafter sees 5 x 53 = 265 unique cards in a full draft pod, compared to 276 for a regular 3 x 15 draft. Do note that the total number of cards needed to construct packs rises from 360 (3 x 15 x 8; number of packs x pack size x number of players) to 440 (5 x 11 x 8), so for those with a big enough cube to accommodate this number, this actually increases the number of cards in your cube that end up in the draft, despite each player seeing slightly less unique cards over the course of a draft.

With 6 players, each player sees 9 + 7 + 6 + 5 + 4 + 3 = 34 unique cards per booster opened. With 7 cards drafted per pack, you want to crack 6 or 7 packs for a 42 or 49 card pool, meaning each drafter sees either 6 x 34 = 204 or 7 x 34 = 238 unique cards in a 6 person draft pod. For this pod size you need 324 (6 x 9 x 6) or 376 (7 x 9 x 6) cards in total versus 264 (4 x 11 x 6) for the 4 x 11 (remember, 2n - 1 cards in a pack) pack size I had been using for a 6 player draft so far.

With 4 players, each player sees 7 + 5 + 4 + 3 = 19 unique cards per booster opened. With 5 cards drafted per pack, you want to crack 9 packs for a 45 card pool, meaning each drafter sees 9 x 19 = 171 unique cards in a half filled draft pod. This pod size uses 252 (9 x 7 x 4) cards versus 180 (5 x 9 x 4) cards for the more commonly (in a 4 player draft) used 5 x 9 packs, or 360 (9 x 10 x 4) cards for the four player format tenchester.

Without having tested it, I like that my suggestion puts more total cards from the cube in the pod's hands. Even though each individual drafter sees slightly less unique cards, my intuition is that this ends up supporting niche archetypes better than the regular 3 x 15 draft.
 
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