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The results of these experiments hinted that there is a chance, though a very small one, to get LD as early as round 4. We failed to get any divinium on rounds 1-3, though it is worth noting that we didn't naturally have much points to spend so early, and we only did a few test games where we left that first room on rounds 1-3. Round 7 seems to be some sort of turning point where the likelihood of getting an LD increases significantly from round 6, and on round 10 we both got an LD usually withing the first couple of door purchases.
Our conclusion was that there's a secret algorithm that works something like this: every time you purchase something, there is a certain likelihood of getting one LD. That likelihood is very small on round 1, it increases every time the round changes, and when you get an LD, it drops back to the value it was on round 1, and starts increasing again based on rounds (not time). It's clear that it is not time based, because the normal rate of getting LD (about every 8 rounds or so) does not change even on really high rounds, even past round 100 it stays the same. And needless to say, a single round takes a very long time on such high rounds.
I'm not completely sure whether you can or cannot get more than one LD per round, but at least the likelihood is extremely small. During the hundreds of zombie hours I've had, never have I gotten more than 1 divinium during the same round. There has been extremely rare cases where I've gotten an LD and another on the next round, but usually it takes that 7-8 rounds to get another one. Those daily challenges that award LD do not affect the "normal" LD drop rate or likelihoods. That way you can get two LDs per the same round, getting a normal one and completing the challenge.