The Bouncer of the Last Circle

Following the success of your last paper, you received an invitation to The Last Circle, a private bar for mathematicians and logicians. But the bouncer in front of the only entrance won't let you in no matter what. After observing (and listening to) some others going into the bar, you've learned that:

1 = 7
6 = 7
8 = 7
9 = 7
10 = 7
1,000,000 = 7

So, to no surprise, you are about to come up to the bouncer and answer with lucky number 7, but two more clients arrive and, this time, the answers are different:

20 = 8
30 = 12

You sit down on a nearby bench, confused, but after a bit of thinking, you come up to the bouncer. He simply says, in a deep voice: "18"

Hints:

Begon, weird shapes !

• I'm fairly sure I have an answer given the most recent hint, but it doesn't agree with the given answers for 3 and 7. Are you sure those are correct?
– fljx
Feb 28, 2022 at 13:29
• Numbers, am I right ? Fixing in a minute. Feb 28, 2022 at 13:32
• (Fixed the examples) Feb 28, 2022 at 13:36
• That works now. I've updated my answer.
– fljx
Feb 28, 2022 at 13:40

11

Because:

The answer for N is the number of letters in the name of an N-sided polygon.
1 - monogon - 7
6 - hexagon - 7
8 - octagon - 7
9 - nonagon - 7
10 - decagon - 7
1000000 - megagon - 7

20 - icosagon - 8
30 - triacontagon - 12

So we have 18 - octadecagon - 11 letters

• This answer should be a comment, really.
– Bass
Feb 23, 2022 at 23:19
• Congrats! And again, deeply sorry for the scuffed examples, numbers are hard ! Feb 28, 2022 at 13:40