3 thoughts on “Group 1 Conjecture

  1. I am proving Samuel Wong’s conjecture. I will slightly change his conjecture and sate it as, “If any shape with an even amount of edges is solvable and we insert interior edges connecting two opposing vertices, that shape is solvable only if the amount of inserted edges are odd. It is not solvable if the amount of inserted edges are even.”

    I began experimenting with this conjecture by trying out some examples. I will attempt to display these below. I will use letters to represent the vertices.

    1) _ a _ _ a _
    | | |_ b_|
    . connecting the two vertices.
    Here both of these are solvable and follow the conjecture.
    2) a__b a_b
    | | c_d
    | X | f, b->e, or c->d, we have a solution. If we have two of these edges present, there is still a solution. If we have three of them present, there is no solution. This disproves our conjecture.

    While I was working on this conjecture, originally I thought it was true. As I started with my examples I was rather pleased. Once I reached a hexagon and discovered the conjecture was false I remained in a good mood. The conjecture was interesting, since it was relatively simple but did not seem obviously true or false.
    While working on these problems I have tried both simple and complex conjectures. While solving these diagrams are relatively easy, developing theorems for them is proving quite difficult, yet still entertaining. I enjoy the challenge and am looking forward to continuing this project.

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