Lady C gives 11 different numbers one of which is her age to logicians A and B. What is the age of lady C?

Question

Logicians A and B ask lady C her age.
Lady C gives 11 different numbers one of which is her age to logicians A and B:
35, 36, 38, 42, 45, 46, 51, 55, 57, 61, 62
Then lady C tells the digit in tens to logician A and tells the digit in ones to logician B.
Logician A says: "I don't know the age of lady C. I think/know logician B does not know either."
Logician B says: "Initially I don't know the age of lady C, but now I know."
Logician A says: "Oh. Then now I know too."

Question: What is the age of lady C?

Answer 1

Lady C is

61

Because

We can see right away that A was told either 4 or 6 for the tens place, since A knows that B doesn't know. This can only happen if all of the options for A have non-unique ones digits. 38 and 57 eliminate 3 and 5 for the tens digit, leaving 4 and 6.

Now

B can deduce this, and this knowledge is enough to pinpoint the number. That means B's number cannot be 2, because 42 and 62 are both options. So we have narrowed the options down to 45, 46, and 61. Whichever of these it is, B can now deduce the correct one.

However,

because A can also deduce the correct age now, it has to be 61. If A's number had been 4, A would not be able to differentiate between 45 and 46. So A must have been told 6, and 61 is the correct age.