All posts by gibson

The Prime Derivative

Given a positive integer \(x\), define the derivative of \(x\), denoted, \(x’\) as:

\(x’ = 1\) if \(x\) is a prime.

\(x’={a}\cdot{b’} + {a’}\cdot{b}\) if \(x={a}\cdot{b}\) (\(x\) is composite)

\(x’=0\) if \(x=1\)

Since all composites are products of primes, we can find the derivatives of all orders for all integers.  What patterns are observed?

Chomp the Graph

Given a finite undirected graph \(G\), players alternate turns and remove either a single edge or a vertex with all incident edges.  Whoever removes the last vertex, leaving their opponent with the empty graph, wins.   Which player has a winning strategy for games on trees, forests, cycle graphs, complete graphs, etc?

diagram

Positive Triangle Game

Two players take turns marking the edges of a complete graph \(K_n\) , for some \(n\), with + or -signs.   The two players can choose either mark (this is known as a choice game).   In Positive Triangle, the first player to complete a triangle with an even number of – signs is the winner.  In this game, the goal or winning triangle can contain marks made by both players.

  1. Under what conditions does the first player win?
  2. If the first player to create a positive triangle loses, under what conditions does the first player win?
  3. For which complete graphs is a draw (no winner) possible

positivetriange