Can’t get my (lug) nuts off

The previous owner swapped lug bolts for studs on my '05 MINI. Is be fine with that if:

a - he used regular lug nuts 

or 

b - he hadn't lost the key for the nuts. 

Other than hammering a 12-point socket over them and spinning them off, are they other options? 
I went to a tire shop and they didn't have a key. I'd rather buy the adapter, but I'll do what I have to do now. 

In reply to DrBoost :

5 grooves, equally spaced?

What about getting a hex socket and removing the studs with nuts included. Easy peasy 

$7.16

https://smile.amazon.com/Point-Spline-Drive-Socket-Locking/dp/B01B3KTGL4/ref=sr_1_2?crid=2OUYI7INJ7ZCZ&keywords=6+groove+lug+nut+key&qid=1644103644&sprefix=6+groove+lugnut+key%2Caps%2C222&sr=8-2

Looks like a spline drive.  I'm surprised the tire shop didn't have one.   Auto Zone/Orielys/etc should have a socket that fits.

I have three of them on my toolbox now but it's a bit far away. 

Dorman makes 6 spline nuts and sockets. Any parts store should have a Dorman catalog and be able to get a socket overnight.

I've got an old Craftsman lug out set I use for old lug locks. I'm sure some other company makes something similar. 

Any other tires shops in the area? I'm surprised the shop didn't have it. A local shop helped me get nuts off my ambulance. 

I have many sets of similar spline drive lug nuts.  A regular socket actually will work on them, I think 17mm.  Might be 16mm.  Definitely needs to be six point, though!