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The reasoning for how we know this:

If A is normal, then we can get a yes or no for the first answer no matter what $S$ is. Because we don't know anything about A's identity after the first question, the answer we get for the $S$ we ask about must be something that either a knight or knave could have said under some circumstance. Thus, the answer to the second question must allow us to distinguish between one case where A is a knight, one case where B is a knave, and the two cases where A is normal.

For the (K)night, (N)ormal, and Knave (sometimes called a (J)oker), we get six possible arrangements and the following possible answers to the second question:

Arrangement | Q2 answer
K N J       | K N J
K J N       | K   J
J N K       | K N J
J K N       |   N  
N J K       |   N J
N K J       |     J

From this we can see that if the answer to the second question is that C is a knave or normal, we won't be able to tell if A was a normal or not. So the answer to the second question must be that C is a knight. Either arrangement with A being a knight allows for that answer, so in order for "C is a knight" to distinguish between A being a knight or knave, we must have narrowed the possibilities down to one of the possibilities where A is a knight, the possibilities where A is normal, and the possibility where A is a knave and B is a knight. So "C is a knight" will always rule out A being normal and B being a knight, leaving A to be the knight.

Here's one possible solution:

$S$ is

the set of the two possibilities where B is normal. In other words, it is equivalent to asking "is B normal?"

When asked if $S$ contains the correct possibility

A said yes. This could mean A is normal, or that A is a knight and B is normal, or that A is a knave and B is the knight.

When asked about C, B said

C is a knight.

Using this, we can determine the identities of A, B, and C:

Neither the knight nor the knave would claim the other is a knight, so A is not normal. Based on the response to the first question we know B is either normal or the knight, but the knight would not claim anyone else was the knight so B must be normal. So A is the knight, B is normal, and C is the knave.

Here's another possibility:

$S$ is

the set {(Knight, Normal, Knave), (Knave, Knight, Normal)}.

When asked if $S$ contains the correct possibility:

A said no. This could mean A is normal, or that A is a knight and the first possibility in the set swaps the normal and knave, or that A is a knave and the second possibility in the set is correct.

When asked about C, B said

C is a knight.

Using this, we can determine the identities of A, B, and C:

Neither the knight nor the knave would claim the other is a knight, so A is not normal. A can't be a knave either, because then B would be a knight claiming a normal was a knight. So A is the knight, B is the knave, and C is the normal.

A will always be

a knight

Here's one possible solution for the questions and identities of the three:

$S$ is

the set of the two possibilities where B is normal. In other words, it is equivalent to asking "is B normal?"

When asked if $S$ contains the correct possibility

A said yes. This could mean A is normal, or that A is a knight and B is normal, or that A is a knave and B is the knight.

When asked about C, B said

C is a knight.

Using this, we can determine the identities of A, B, and C:

Neither the knight nor the knave would claim the other is a knight, so A is not normal. Based on the response to the first question we know B is either normal or the knight, but the knight would not claim anyone else was the knight so B must be normal. So A is the knight, B is normal, and C is the knave.

Here's another possibility:

$S$ is

the set {(Knight, Normal, Knave), (Knave, Knight, Normal)}.

When asked if $S$ contains the correct possibility:

A said no. This could mean A is normal, or that A is a knight and the first possibility in the set swaps the normal and knave, or that A is a knave and the second possibility in the set is correct.

When asked about C, B said

C is a knight.

Using this, we can determine the identities of A, B, and C:

Neither the knight nor the knave would claim the other is a knight, so A is not normal. A can't be a knave either, because then B would be a knight claiming a normal was a knight. So A is the knight, B is the knave, and C is the normal.

Here's one possible solution:

$S$ is

the set of the two possibilities where B is normal. In other words, it is equivalent to asking "is B normal?"

When asked if $S$ contains the correct possibility

A said yes. This could mean A is normal, or that A is a knight and B is normal, or that A is a knave and B is the knight.

When asked about C, B said

C is a knight.

Using this, we can determine the identities of A, B, and C:

Neither the knight nor the knave would claim the other is a knight, so A is not normal. Based on the response to the first question we know B is either normal or the knight, but the knight would not claim anyone else was the knight so B must be normal. So A is the knight, B is normal, and C is the knave.

Here's one possible solution:

$S$ is

the set of the two possibilities where B is normal. In other words, it is equivalent to asking "is B normal?"

When asked if $S$ contains the correct possibility

A said yes. This could mean A is normal, or that A is a knight and B is normal, or that A is a knave and B is the knight.

When asked about C, B said

C is a knight.

Using this, we can determine the identities of A, B, and C:

Neither the knight nor the knave would claim the other is a knight, so A is not normal. Based on the response to the first question we know B is either normal or the knight, but the knight would not claim anyone else was the knight so B must be normal. So A is the knight, B is normal, and C is the knave.

Here's another possibility:

$S$ is

the set {(Knight, Normal, Knave), (Knave, Knight, Normal)}.

When asked if $S$ contains the correct possibility:

A said no. This could mean A is normal, or that A is a knight and the first possibility in the set swaps the normal and knave, or that A is a knave and the second possibility in the set is correct.

When asked about C, B said

C is a knight.

Using this, we can determine the identities of A, B, and C:

Neither the knight nor the knave would claim the other is a knight, so A is not normal. A can't be a knave either, because then B would be a knight claiming a normal was a knight. So A is the knight, B is the knave, and C is the normal.