I wrote PT53 yesterday and spent probably 10 of 35 minutes on question 22. My setup for the game was awful, and I redid it with a different setup (seen above) in 5:45. I originally tried to use FGH as the base, but above I used 1 2 3 as the base with much greater success.
F/G and H/F signify an implied labeling. I recognized that those were the possibilities, but didn't bother to write them down in my main diagram because I didn't want to clutter it up.
The ovals represent what I call "soft blocs" in that their order doesn't matter, so long as they are vertically aligned. The soft not-bloc is similar in that no matter the order, they cannot be vertically aligned.
The P ---- N and G ---- H are diagramming elements from a sequencing/relative ordering game, whereby it simply shows that P must precede N, and G must precede H, whether they are 1,3 2,3 or 1,3. I always make sure to elongate those lines so I don't confuse them with P_N or something that might indicate there must be some space between them, since they could occur concurrently.
Now, onto the questions:
18 is a rules check. Apply the rules to find that E is the correct answer. A violates rule 2. B violates rule 3. C violates rule 4. D violates a combination of rules 1 and 5 (because G can only be first or second, S cannot be on the third place team).
19 is solved with a hypothetical diagram (figure 1.1). A is shown to be possible. If P is on H, G must be first and includes S. P must precede N, so N must be third and on team F. M and O remain undetermined.
20 is solved with a hypothetical diagram (figure 1.2). A is not possible from the diagram, but B is possible. If O is on the second place team, T must be his teammate. P must both precede N and not be teammates with M, meaning his only possible teammate is S, who must be on team G.
21 is solved by 19's hypothetical diagram (figure 1.1). S, T, P, N are determined.
22 is solved by a double hypothetical (figure 1.3) because M can only be first or second place if it is to place higher than team H. F can only place second or third, so A is out. G must place first, so B is out. N must place third, so C is out. O must also place third, so D is out. P can be first given the first hypothetical.
23 is partially solved by the previous hypotheticals. Figure 1.1 eliminates A and C. Figure 1.2 eliminates D. Figure 1.4 eliminates E, leaving D as the only remaining possibility.
======================
Possibly of note are the circles around certain variables in the hypotheticals above. Whenever I diagram a hypothetical, I always place a circle around any catalyzing variables. If the initial condition(s) determine(s) the diagram, knowing that can be of greater value if tested on it in a later question. For example, knowing that S is on team 2 completely determines the diagram. This didn't serve as useful information in this particular game, but it often does.
If there were a question 24 which asked "If S is second, which could be true?" I would know that I can use that hypothetical exclusively to answer the question. If I didn't mark S as determining the diagram, I may take time to think up additional hypotheticals.
Consequently, if hypothetical question 24 were to ask "If M is third, which of the following could be true?" I know that my hypothetical is limited in its ability to answer this question.