Each mini-puzzle is used to form a graph. The first puzzle forms the reference graph, which has 8 vertices and 13 edges. Each other graph has 8 vertices and 12 edges, because each other graph is missing exactly one edge from the reference graph. To solve the puzzle, find the missing edge from each graph that's not in the reference graph, and take the first letter of the two vertices that it connects. The results that you get are 'GU', 'UN', 'NS', 'SH', 'HO', 'OT'. String them together to get GUNSHOT, which is the solution to the puzzle.
Here is a picture of the reference graph (the numbers are arbitrary, but used in the rest of this file to order the nodes):
The graphs are formed in the following ways:
Reference graph: Build the six pieces into a dodecahedron.
Upgrades picture: This is an adjacency matrix representation of the graph. Each letter is a vertex, and an 'X' indicates a connection between the row and column vertices. The missing edge is between the U vertex and the G vertex, yielding 'GU'. In order of the reference graph, the vertices are:
Years: The vertices are countries, and the edges are wars fought between them that take place during the given years. The missing edge is between Norway and the United States, yielding 'UN'. In order of the reference graph, the vertices are:
Numbers: The vertices are the numbers given, and the edges are prime factors of the given numbers. The missing edge is between Ninety-One and Seven Thousand Nine Hundred and Five, yielding 'NS'. In order of the reference graph, the vertices are:
Dating Puzzle: The people are the vertices, and there are edges between any two people that have ever dated. The missing edge is between Haley and Sidney, yielding 'SH'. In order of the reference graph, the vertices are:
Story: Each place is a vertex. The edges mean that the protagonist traveled directly from one place to another. The missing edge is between Oak Square and Harvard Square, yielding 'HO'. In order of the reference graph, the vertices are:
The Stata Center
Kendall/MIT T Stop
77 Mass Ave
Words: The words are the vertices and the edges are 2-letter substrings in common between words. The missing edge is between operand and tinge, yielding 'OT'. In order of the reference graph, the vertices are:
2006 MIT Mystery Hunt