The odds are brutal. For a given age there are 18 unique blueprints. Aid/Polish/Motivate will get you a blueprint of that age about 1% of the time, but you can also try for Great Building award packages, or spend 200 diamonds to get exactly the blueprint that you need.
If you can use 7 of the 18 possible blueprints, then it's obvious that the odds of your NEXT blueprint being useful are 7:18 but it's more convenient to think in terms of how many draws you'll need to get any one of the remaining useful blueprints and that's 18/7 draws.
# | Draws |
18 | 1.000 |
17 | 1.059 |
16 | 1.125 |
15 | 1.200 |
14 | 1.286 |
13 | 1.385 |
12 | 1.500 |
11 | 1.636 |
10 | 1.800 |
9 | 2.000 |
8 | 2.250 |
7 | 2.571 |
6 | 3.000 |
5 | 3.600 |
4 | 4.500 |
3 | 6.000 |
2 | 9.000 |
1 | 18.000 |
| 62.912 |
If you want either of the 2 Great Buildings, then just ignore the first 9 terms.
So if you have average luck,
- you'll need 63 blueprints for BOTH buildings in each age
- you'll need 51 blueprints for your first building in each age
Assuming that you're visiting 50 cities every day, and carefully selecting the target age, that's one blueprint every 2 days. It certainly SEEMS that the yield is higher, but every time I actually count it's right at 1%. So you'll need most of 4 months for your first Great Building, if you just depend on Aid/Polish/Motivate.
You can also improve your odds just a bit by converting 2+2 or 3 duplicates into a random blueprint for that particular Great Building, which is the equivalent of two draws in the above table, so it's not a very good strategy as you'll need those blueprints once you max out your Great Building at level 10.