Does a maximally clumped deck or a mana woven deck require require more shuffling to sufficiently randomize?

Question

I've been under the impression that a mana woven deck (i.e., nonland nonland land nonland nonland land...) reaches a state of randomization after fewer riffle shuffles than the same deck with all nonlands on the top and all lands on the bottom. More generally, the idea would be that more clumping requires more shuffling to achieve sufficient randomness.

Recently, however, I've heard it claimed that a certain style of weaving (or double pile shuffling) requires twenty riffle shuffles to make random. This struck me as odd, since 7-9 riffle shuffles are generally considered sufficient.

Do clumped or declumped decks require more shuffling to randomize, or equivalently, given a fixed number of riffle shuffles, is one more likely to end up with a sufficiently randomized deck starting with a clumped or declumped initial distribution of cards?

Answer 1

It doesn't make a difference.

No deck in a known order is more or less random than any other deck in a known order. A deck with all of the lands together requires just as many shuffles for randomization as a deck with the lands perfectly distributed. A mana weave starts with looking at the cards to see which are lands and which are not, and doing so resets the deck to a "known order" state and undoes any previous shuffles.

To see why this is true, we can characterize the degree of randomization of a deck by the number of possible permutations it can possibly be in. Any deck with a known order has one possible permutation, and a fully shuffled deck can have every possible permutation. A partially shuffled deck is somewhere in between.

A pile shuffle deterministically maps any given permutation to a specific other permutation, so it changes the set of possible permutations, but always preserves the size, and therefore preserves the degree of randomization.

If you experience that reordering your cards before shuffling (such as by mana weaving) results in a noticeable change in your draws, then you are not shuffling enough.

Answer 2

murgatroid99's answer sufficiently answers the question, this is more of a philosophical take on deck randomisation.

A perfectly random ordering is such that every possible order is equally likely. This is exactly equivalent to each position containing a uniformly random card, or each card being in a uniformly random position.

Another important factor to shuffling a deck, is the amount of knowledge you have about the order of the deck. Eg. If you somehow perfectly shuffle a deck then look at it, from your perspective the number of possible permutations went from N! to 1.

Does this mean that just leaving your deck as is, would be random? Kind of... It's a bit like looking the face of a dice from the last game you played instead of rolling it, but a deck of cards has more possibilities, therefore more room for unintentional ordering. From last time you played, your lands and creatures are probably in clumps. Not to mention, you may even remember what you had last game, and where they are in your deck. Shuffling is a way to mitigate this.

One way to perfectly shuffle a deck, (although time consuming) would be to have another person take your deck face-down, select a card at random (using a dice or other RNG), and put it in a new pile. They repeat this until every card is in the new pile. This satisfies every aspect of being perfectly random; each card could be anywhere, and you know nothing.

Obviously, this is impractical to do, so you want something as close as possible. Multiple (7-9) riffle shuffles are usually sufficient to come close enough to uniform randomness, and to obfuscate your memory of card positions. Since this is the physical world, there are some caveats. It depends on a certain amount of dexterity. If your riffles consistently leave the top or bottom cards in the same position, a few overhand shuffles between riffles should suffice. Or, if your cards are prone to sticking together while riffling, one pile shuffle (without looking!) can separate them again. Note that overhand or pile shuffles alone are generally considered insufficient forms of shuffling.

Pile shuffling alone, deterministically rearranges the deck, thus not increasing randomisation. Equivalently, it's flipping a dice onto its opposite face instead of rolling it.

Mana weaving is more detrimental. It's actively arranging the deck according to a rule. It can never increase the randomisation. Moreover, it introduces information to you about the order. If not specific card places, you now know that after drawing 2 nonlands, the top card is likely a land. It's the equivalent of turning your dice to a face you want instead of rolling.

A perfectly shuffled deck is very unlikely to have a neat pattern of land/nonland cards. Just like how a standard deck of playing cards is unlikely to be shuffled in order of black, red, black, red, etc. You should expect some clumping even in a perfectly shuffled deck.