What Programming methodology is this?

It seems like a team building exercise. Everyone sees how everyone else works, and helps them grow. The 'client' is there to provide feature requests to see how whoever's on the hot-seat implements those and responds to changes/criticisms.

Note: In this scenario the client might just be a senior programmer, pretending.

Mobbing?

http://benjiweber.co.uk/blog/2015/04/17/modern-extreme-progr...

Probably "Cheap programming" as you can't afford to have more than one computer, everyone gets chance on that one computer in round robin way.

It's the "Convince-your-competitors-to-use-it" methodology.

Burnout driven programming.

search google for "pair programming"

Well pair programming is when there are two programmers working together on one terminal.

When you have the whole team waiting behind the guy at the keyboard, I would call this methodology "Dumbest".

extreme programming ?