I'm late to the party but did anyone ever try this method? All you need one of each polyhedral dice (d20, d12, d10, d8, d6, d4)
When the game tells you to pull from the tower, roll the d20. If you roll a 1 or 2, replace the d20 with the next biggest die (d12). When you roll 1 or 2 on the d4, the tower fall.
According to my tests, the average number of pull is around 30 and the probabilities are very similar to other methods. Plus you only need 6 common dice.
EDIT: I just realized that this is not a great method since there's no immediate sense of danger as long as the die is not a d4.