I've been learning more about sapere. I found the following resource but I'm missing a subtle difference between "sa" and "sai". These quotes are from a university so I assume they are correct but want to understand why.
http://www.uky.edu/~allaire/CONSAP.htm in the answer key under "II. Sapere..." you'll see 2 examples. They have:
"Mi scusi, signore, sa per caso che ora è?"
"Scusi, signora, sa se l'autobus numero 27 si ferma qui?"
In both cases above, instead of "sa" why not use "sai" which literally means "you know"? like
"Mi scusi, signore, sai per caso che ora è?"
The only thing I can think of is, in this formal context "sa" might be "Lei sa"?