Okay, maybe this will help. Here's a *ONE4 itinerary we've considered, which is "legal" according to Mileage Monkey and uses up our 20 segments:

DEN-LAX-SFO-SYD-ADL-BNE-PER-NRT,KIX-HKG-SIN-BKK-DXB-LHR-DME-MAD-LHR-YVR-JFK-MIA-DEN

42,488 miles

However, it only uses 3 of 4 possible legs in Asia and South West Pacific. The LAX-SFO and LHR-YVR legs could be dropped if necessary. They are pretty much "filler" to add miles and legs.

Overall the destinations are ones we'd be happy with, but I'm pretty sure this isn't the optimal itinerary in terms of getting us on the best flights as well as maximum mileage. Suggestions?