Mathematical Theories
Number of Perfect Squares in Range Every natural number has a unique square. So the number of perfect squares between 1 and x would be equal to the biggest positive integer k whose square would be less than/equal to x. k = floor of square root of x. Eulerian Path An Eulerian path on a graph is a traversal of the graph that passes through each edge exactly once. It is an Eulerian circuit if it starts and ends at the same vertex. Theory for being Eulerian Path: If a graph admits an Eulerian path, then there are either or vertices with odd degree. If a graph admits an Eulerian circuit, then there are vertices with odd degree. Hamiltonian Path A Hamiltonian path is a traversal of a (finite) graph that touches each vertex exactly once. If the start and end of the path are neighbors (i.e. share a common edge), the path can be extended to a cycle called a Hamiltonian cycle . Theory for Hamiltonian Cycle Dirac: A simple graph (one without multiple