My feeling is that it’s an error. The only way to get √(9!)=6 is to write it as √(9)!=6. That means moving the factorial symbol out from under the vinculum.
Or is there some bizarre rule pertaining to the factorial operation (which, as far as I know, is just multiplication) that makes it acceptable to write it this way?
You’re right, the clock is in error: The only rule of mathematical notation is clarity, and that is quite clearly wrong. There’s no special mathematical rule which makes the square root of nine factorial equal to six.
Well they used a vinculum, which is a form of grouping just like parentheses (to show the radicand whose root is being indicated). The only reason I substituted parentheses was because I didn’t know how to create a vinculum here.
The whole thing even seems more contrived than usual for such clocks. Most positions seems to be little more than x/x, (x+x)/x and so on. Apart from perhaps selling well to adherents of the Bahá’í faith I can’t see the attraction.
It seems to me that you also need to move the vinculum to be inside the parentheses, like this: (√9)!
Now it unambiguously means “three factorial”. When the “9” is inside the parentheses all alone, the parentheses don’t really clarify anything.
Actually, the line is over both the decimal point and the nine, so the implication is really that .9.9.9.9.9.9… = 1. As far as I know there’s no standard mathematical notation which uses multiple decimal points in this manner, so I think this is yet another error in the clock.