Which Alice?

Solutions to the Puzzles
and that A is the spy; if Mr. Anthony answered negatively, then the
friend would have been unable to know which of Cases 1b, 2a, 2b
held, and could not have known whether A or B was the spy. So the
only way that the first friend could have solved the problem is that
Mr. Anthony answered affirmatively and that Case la holds.
As to the second friend, if Mr. Anthony answered him affirmatively,
then this friend would have known that Case 2a must hold
and that A is the spy, but if Mr. Anthony answered him negatively,
then the second friend couldn't have solved the problem. And so
the only way the second friend could have solved the problem is
that Case 2a holds and that Mr. Anthony answered him affirmatively.
Now it cannot be that Case la and Case 2a both hold, and
therefore Mr. Anthony could not have given an affirmative answer
to both his friends, and so it is impossible that both friends were
able to solve the problem. Therefore neither friend solved the
problem (since we are given that either both did or neither did),
and so Mr. Anthony did not answer either question affirmatively.
This rules out Case la and Case 2a, so B must be the spy.
Chapter 6
THE FIRST QUESTION Alice made the mistake of writing
eleven thousand eleven hundred and eleven as 11,111—which is
wrong! 11,111 is eleven thousand, one hundred and eleven! To see
the correct way of writing eleven thousand eleven hundred and
eleven, add them up like this:
So, eleven thousand eleven hundred and eleven is 12,111—which
is exactly divisible by 3.