I don't have the maths to work this out, but what does the curve look like if you draw a random domino from a full set of 28 and continue until you get a double, removing dominos after they're drawn?
This is the curve if you stop once you've pulled every double—if you only need to draw one then it's like rolling a 1 on a d4 (since there are 7 doubles in the total 28), which ends very quickly. Pulling all works pretty well, with some specific features:
- it always ends after turn 28 at most
- 28 = four 7-day weeks (plus a possible 28 Days/Weeks Later reference in particular if that's what you're going for?)
4 years ago(+1)(-2)
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For real though thanks for running the maths on that - I wouldn't know where to start!