# Data analysis: How did people beat the Great Hall game?

This is the game:

There is a Great Hall with 102 doors. 100 of these doors lead to one of 100 different side rooms. The 101st door, at the end of the Great Hall leads to the Great Tower, where unspeakable pleasures await. The 102nd door, at the other end, leads to the outside, where the ordinary world and its doldrums and tribulations await.

The object of the game is to exit the Great Hall to the Great Tower. To exit through the door, one must solve a riddle. The riddle cannot be solved without the player reading at least one clue. Each side room has a clue written on the wall. Each side room has a name that vaguely explains the name of its clue, but is not itself enough to be useful to solve the riddle; one must actually enter the room and read the full clue. Each side room also has a list of some of the other rooms and their names.

To play the game, a player picks which room they will start in based on the name of the room (and how useful they think the clue will be). They are then dropped through a hole into that room, where they can read the clue. After they read the clue, they can look at the list of other rooms - we don't know which other room(s) is/are listed in any given room - and decide which room they will visit next... provided the clue they just read did not allow them to solve the riddle and enter the Great Tower. The player can then walk into the Great Hall to the next side room, and read that clue in the hopes that the sum total of the clues they have learned will help them solve the riddle.

Millions of people play the game, all one at a time. They come in all types. Some are absolutely determined to solve the riddle as fast as possible and they manage to solve it with just one clue, and enter the Great Tower. Others are minimally motivated and give up easily, walking out into the ordinary world. Some players may find that a clue or string of clues is so confusing that it overwhelms them and they give up in frustration no matter how determined they were, exiting into the ordinary world. Some player backtrack to previous rooms to re-read clues.

Analyzing the game:

You receive a list, a record of every player who entered the Great Hall, and the path they took, every room they visited or re-visited, and in what order. You must use this record to piece together the various ways that players beat the Great Hall game, that is, which side rooms are the most popular first rooms, and what combinations of rooms/clues most commonly lead to solving the riddle.

Analyzing the information, you find that 8 of the 100 rooms are most correlated with solving the riddle (the 9th most correlated room was an order of magnitude lower and thus you discard it and the rest). You now want to know, are all 8 of these rooms correlated with one another? Or are there a couple clue-sets whose rooms may or may not overlap? In particular, you want to know with as much precision as possible, what is the most powerful combination of rooms in terms of helping players solve the riddle.

With such a large number of players, every combination of those 8 (or fewer) rooms occurred at least once.

How do you figure out what combinations of those 8 rooms are linked to each other?

How do you strike a balance between a very specific set of these rooms- e.g. any player who visits all 8 - which would occur less frequently than, or a less specific set of rooms (any player who visits a specific three-set of those rooms but might also visit any of the other 8) which would occur more frequently but be less descriptive and offer an incomplete picture.

To be more concrete, imagine the eight rooms are labeled A, B, C, D, E, F, G, and H. Perhaps it is very common for players to go to rooms A, C, D, and E. Any query that includes [A,C] or [C,D,E] would include [A,C,D,E]. But obviously [A,C] or [C,D,E] will turn up more results.

If you were to rank all possible combinations of these rooms* for correlation to whether they solve the riddle, the highest ranked ones might be more specific (more of the 8 rooms required) but possibly less representative and edge cases. If you ranked them by frequency, obviously the less specific ones would be at the top.

*IOW game sessions/players who visited that combination of rooms.

How does one find the perfect combination of rooms that is both highly correlated to solving the riddle and also common in the data? What if I want to find different, minimally overlapping sets of those 8 (perhaps [A,B,C], [D,E,F], and [F,G,H] are distinct sets.

I'm pretty clueless with math so help me out!

Don't hesitate to ask for clarification.