Just so we’re straight, each row represents one of the four houses. The first column, Column H, is just the name of the house (type G, H, R, S). The first column does not contain any people, and that column is really not considered part of the proper puzzle. Nothing is adjacent to the H column and nothing shares diagonals with the H column. In other words, A1 is a corner cell and it only has three adjacents.
2/24) For four people to share a diagonal, they all must be in a different house. Here are the possibilities: A4, B3, C2, D1 OR A1, B2, C3, D4 OR B1, C2, D3, E4. Hagrid also says that only one of those is adjacent to Harry. In the first option two of them are adjacent to Harry: A4 and B3. In the third option none of them is adjacent to Harry. So the four characters Hagrid mentions are in A1, B2, C3, D4. But you know which one Ron is, since he is adjacent to Harry. C3 is Ron.
3/24) If Harry, Ron, Fred, and George are all in a different house, Fred and George are each in one of the top two houses. Since neither is adjacent to Ron and Hagrid is in E2, you know that one of the twins must be in A2. Harry says that a professor is adjacent to Hagrid and Fred. No one could be adjacent to both Hagrid and A2, so it must be George in A2.
4/24) George says that the top row is not Gryffindor and tells you that B1 is Molly.
5/24) Hagrid tells you that Moody is either A1, B2, or D4. Molly tells you that Moody is not adjacent to anyone in Ron’s year. This means Moody must be in A1.
6/24) Harry tells you that Hermione and Ginny are somewhere in the top two houses, one each. Ron tells you that Fred is in the top house. So the top house contains Hermione/Ginny and Fred. Moody tells you that C1 is neither of those, meaning they are in D1 and E1. This means the professor adjacent to both Hagrid and Fred (Harry’s clue) must be in D2. Harry also tells you this professor was a Defense Against the Dark Arts teacher. You know that B2 is Remus or Albus (Hagrid’s clue), so Moody tells you that C2 (Hermione/Ginny) and D2 (that professor) are both female. The only female DADA professor Harry had is Umbridge. She is D2.
7/24) Let’s recap what you know: Top row is not Gryffindor and contains Fred and Ginny/Hermione in D1 and E1. C1 is some older non teacher or some young person or a muggle.
House 2 could be anything and contains Lupin/Dumbledore in B2 and Ginny/Hermione in C2.
House 3 – who knows?
House 4 contains Lupin/Dumbledore in D4.
You also know that Draco and Voldemort are in the quiz, in Slytherin, which could still be any row. Actually, it couldn’t be House 1 because you know who goes in the empty cells except C1. Slytherin can’t be House 2 because you have narrowed down its empty cells too. Slytherin can’t be House 4 because D4 is Lupin/Dumbledore. Tom (or Lord or Voldemort) and Draco both contain ‘O,’ so neither of them can be in a corner, because “most” out of four is three, and Alastor already is in a corner. At least one of Draco/Tom would need to be in a corner in order for Slytherin to be House 4. Therefore House 3 is Slytherin. Put S in H3.
8/24) Hermione is either C2 or D1 or E1. Slytherin tells you she is not D1 because you know that D4 is Professor Lupin or Dumbledore. But Hermione was also ruled out of E1 by Umbridge, since her name contains an ‘O.’ Therefore Hermione is C2.
9/24) It’s important to realize that you now essentially know all the characters in the quiz. There are 12 missing spaces and here’s whom you haven’t placed but know is in the quiz: Snape (Molly), Lupin (Molly/Hagrid), Voldemort and Draco (George), James/Sirius/Wormtail, Vernon, and Arthur (Hermione), Dumbledore (Hagrid), Neville and McGonagall (Slytherin), Fred (Ron), and Ginny (Harry).
Now Hermione tells you the marauders are adjacent. Lupin is either B2 or D4. If he’s in D4, he can’t be adjacent to the other marauder. This is because of who can possibly go in C1. It can’t be Snape according to A1. Draco and Voldemort are in Slytherin. Neville and McGonagall are in the same house; there’s no room for them in the top house. Dumbledore is B2 if Lupin is D4. Fred and Ginny are somewhere in D1 and E1, not C1. So if Lupin is in D4 and the other marauder is adjacent to him and not in C1, C1 would have to be Vernon or Arthur, and Hermione says they can’t be adjacent to Molly. Lupin in D4 does not work. Lupin is B2 and the other marauder is C1.
10/24) Hagrid tells you that Dumbledore’s in D4.
11/24) B2 gives you an either/or. If every surviving male Weasley (George, Ron, Arthur) is adjacent to another surviving Weasley, Arthur would have to go in B3. That would mean Slytherin has no room for Neville and McGonagall, who must be in the same house. So they would go in the bottom house. Vernon and Snape would be left over for the bottom two houses. No big deal. BUT WAIT! If Arthur is in B3, where could Vernon go? Hermione says they’re not adjacent and Dolores says Vernon can’t go in a corner. If Arthur is at B3, Vernon must be in Slytherin, which means we have to consider Dumbledore’s either/or as well. If all surviving male Weasleys are adjacent to another surviving male Weasley, then Arthur is in B3. If Arthur is in B3, Vernon cannot go in the bottom house. Vernon cannot go in House 1 either, because Fred and Ginny are up there, and Hermione says Vernon’s not adjacent to Molly. So if Arthur is in B3, Vernon has to be in Slytherin. Dumbledore says if Vernon’s not in Gryffindor, Harry is adjacent to three students and McGonagall. So Arthur in B3, Vernon in D3 or E3, Draco in A3, and Neville/McGonagall in A4/C4. That all works fine EXCEPT where Molly says Snape has to be adjacent to someone from Ron’s year. That scenario would force Vernon and Voldemort into D3 and E3, thus Snape into E4, not adjacent to any students at all. So the entire thing doesn’t work, meaning McGonagall is in E3.
12/24) Harry is clearly not adjacent to McGonagall, so Vernon is in Gryffindor. Now that McGonagall is in Slytherin, Neville must also be in Slytherin. You know from George that Slytherin also contains Draco and Voldemort, so Vernon can only go in the bottom house. So the bottom house is Gryffindor.
12/24) Vernon has to be in the bottom house, since Slytherin still needs Draco, Voldemort, and Neville. So according to Umbridge, Vernon is C4.
14/24) Vernon mentions Harry’s godfather. So now you know the last person in the quiz is not James or Wormtail, but Sirius. And according to whom you need in Slytherin and according to Hermione, Sirius goes in C1.
14/24) Sirius was either Ravenclaw or Hufflepuff, so according to Vernon, H1 is Hufflepuff.
16/24) So H2 is Ravenclaw.
17/24) Slytherin still needs Draco, Voldemort, and Neville. Hufflepuff still needs Fred and Ginny. So Snape must be in Gryffindor. Sirius says Snape is adjacent to Voldemort, so if Snape is at E4, Voldemort would be D3. But again, Snape would not be adjacent to any student, contradicting Molly. So Snape must be A4.
18/24) Slytherin still needs Draco, Voldemort, and Neville. Hufflepuff still needs Fred and Ginny. So Arthur goes in E4.
18/24) Snape does not share a diagonal with a female Weasley. Minerva says he does. So D1 must be Ginny.
20/24) Ron now tells you that Fred is E1.
21/24) Since Voldemort must be adjacent to Snape (Sirius), you know that D3 is Neville or Draco. This means that Hagrid is adjacent to three students. So Fred tells you that Vernon must be adjacent to at least four. So B3 must also be a student, meaning A3 is Lord Voldemort.
22/24) The only married couple who are adjacent to each other is Ron and Hermione. Arthur says they have to be adjacent to Voldemort if Draco is adjacent to Voldemort. That’s not possible, so Draco is not adjacent to Voldemort, but is in D3.
23/24) Neville is all that’s left!