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Speeding up Mana Screw

With the recent legalization of Un-sets in EDH, Mana Screw has become a playable card. This card can theoretically let you suddenly spike ahead on mana. However, if you have a large amount of mana and need to get to another large amount of mana, it can take a lot of flips to actually reach a conclusion. The goal is to find an equivalent process that produces the same results as actually flipping, but without wasting as much time.

There are a few easy assumptions to make for ease of computation and explanation. One, this process ignores colors of mana. It's not hard to extend it to colored spells, but colors are ignored for now. Two, the player has an intended amount of mana to get to, and won't stop until they either reach it or run out. This is how one will optimally play the card, assuming they use it at all, so it's not much of an assumption.

The initial amount of mana is denoted by k and the target amount of mana is denoted by n.

First, we should realize that the odds of getting to n mana are k/n. An easy way to see this is to note that if P(k,n) is the probability of success, P(0,n)=0; P(n,n)=1. Let P(1,n)=x. P(1,n)=0/2+P(2,n)/2=x, so P(2,n)=2x. Similarly, P(3,n)=3x, and so on, so P(n,n)=nx=1, so x=1/n. Thus, P(k,n)=k/n.

If n is public knowledge, then we can simply roll a die with at least n sides, and say that if the result is between 1 and k, we get to n mana; if the result is between k+1 and n, we get 0 mana; if the result is greater than n, we reroll. If you don't care about keeping your spell private, this is a fast way of getting through the flips.

However, one might want to keep the amount of mana that they need hidden. One might be tempted to, at this point, simply write down the target amount of mana, roll the die in private, and then only reveal the result on success (which the other players can verfiy). However, the problem with this protocol is that it doesn't give the other players any idea what n is on a failed attempt. If flipping coins, the other players get to see what numbers you reach, then choose to continue flipping. Thus, we need a different protocol.

The proposed protocol is thus: the player with the mana screw repeatedly tries to get one more mana than they had previously, or lose it all. In other words, they roll a k+1 sided die (or equivalent, as described above). On a 1 through k, they get to k+1 mana. On a k+1, they lose all their mana. They then repeat this process until they hit 0 or n mana.

This process is, on average, faster than simply flipping coins in expectation. This process is O(dlog(n-k)), where d is the size of the die you are rolling, while normal coin flips take k(n-k) flips to terminate in expectation.

Hopefully this makes at least one fewer playgroup tear their hair out, or was at least an interesting read.

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