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I've been trying this challenge for a while and can very confidently say that the Outer Room's draft pool isn't 100% set but rather appears to be on some kind of cycle because I've had the Tomb as a draft option both from the get go as well as after one or two rerolls!
When drafting the outer room as the 10th room I draft in the run (so 9 rooms drafted in the house, then going to draft Outer Room), I got the Tomb in the first set of 3 in the exact same slot and with the other two also matching the other time I drafter the Tomb as the 10th draft of the run previously.
Granted, my data set is currently very small (two times), but getting the exact same draft options is to me a pretty big indicator there's a cycle going. What that cycle is exactly I don't know though.
Two dice will guarantee you'll get the tomb, but it IS possible to draft it with just one or no re-rolls if you get lucky!
Also worth mentioning that if you place the secret garden on the west side, you can actually reach the second lever without needing the power hammer.
Now I have to do the 20 Dead End run :')
A few additionnal comments:
Keys:
Very situational but having the Electro Magnet allows you to steal about 25 keys from the locksmith.
Dice:
About 95% of the time you need 2 dice to get the Tomb but you still have 5% chances to get it with 1 single dice.
Closet can contain 1 dice but IMHO it isn't that usefull because you need 2 most of the time.
You definetely get a chest 100% of the time if you draft it as your first room (middle door)
A couple of times a found 2 dice on the desk of the Office.
For the rooms:
Study should be in Higher tier as it allows you to reroll the room and consequently replace the dices to get the Tomb.
Locker Room & Sauna are available Day One
Although dice are by far the best way to draft the tomb, a study (and sufficient gems) can be used as a substitute if you can't find any.
Draft your dead ends early so they don’t appear later.
Darkroom is not an S-tier room. The step cost of going back to the breaker box can be a deal-breaker in a run
Archive is surprisingly usable if it is drafted late game. The fact that it is a four-way room with no cost can make it a decent option if you don't have many options to pick from. Plus, an experienced player can deduce the archived room from context. For similar reasons, Weight room, isn't that bad of a pick either, especially if you have low step count (< 30). So, I wouldn't actively avoid them, but instead draft them at proper times.
And, probably most important of all, if there is a chance of a successful run, it's still worth going for. My winning run didn’t get a way to re-roll until my last available room slot (It was study).