A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work.
Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved).

a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.
b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
1. Let \( a, b, c, d \) be the number of members of party \( A, B, C, D \), respectively. Initially, we have:
   \[
   a = 31, \quad b = 28, \quad c = 23, \quad d = 19
   \]

2. We claim that \( 3a + 2b - c \pmod{4} \) is constant. To verify this, consider the effect of the wizard's rule: when three members of three different parties start talking, they are converted to the fourth party. This means:
   - Three variables decrease by 1, and the fourth variable increases by 3.

3. Let's check the change in \( 3a + 2b - c \) under this operation:
   - If \( a, b, c \) decrease by 1 and \( d \) increases by 3:
     \[
     3(a-1) + 2(b-1) - (c-1) + 3d = 3a - 3 + 2b - 2 - c + 1 + 3d = 3a + 2b - c + 3d - 4
     \]
     Since \( d \) increases by 3, the term \( 3d \) does not affect the modulo 4 property:
     \[
     3a + 2b - c \equiv 3a + 2b - c \pmod{4}
     \]

4. Similarly, if \( a, b, d \) decrease by 1 and \( c \) increases by 3:
   \[
   3(a-1) + 2(b-1) - (c+3) + 3d = 3a - 3 + 2b - 2 - c - 3 + 3d = 3a + 2b - c + 3d - 8
   \]
   Again, modulo 4:
   \[
   3a + 2b - c \equiv 3a + 2b - c \pmod{4}
   \]

5. Therefore, \( 3a + 2b - c \pmod{4} \) is indeed constant. Initially:
   \[
   3(31) + 2(28) - 23 = 93 + 56 - 23 = 126
   \]
   \[
   126 \equiv 2 \pmod{4}
   \]

6. Now, note that \( a + b + c + d \) is constant and always equal to:
   \[
   31 + 28 + 23 + 19 = 101
   \]

7. Since \( 3a + 2b - c \equiv 2 \pmod{4} \), we need to determine if all members can belong to a single party. Let's check each party:
   - If all members belong to party \( A \):
     \[
     3(101) + 2(0) - 0 = 303 \equiv 3 \pmod{4} \quad \text{(not possible)}
     \]
   - If all members belong to party \( B \):
     \[
     3(0) + 2(101) - 0 = 202 \equiv 2 \pmod{4} \quad \text{(possible)}
     \]
   - If all members belong to party \( C \):
     \[
     3(0) + 2(0) - 101 = -101 \equiv 3 \pmod{4} \quad \text{(not possible)}
     \]
   - If all members belong to party \( D \):
     \[
     3(0) + 2(0) - 0 = 0 \equiv 0 \pmod{4} \quad \text{(not possible)}
     \]

8. Therefore, the only party that is possible for all kidnappers to be in is \( \boxed{B} \).