Mathematical Theories

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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 edges between any pair of vertices, and with no edges that begin and end with the same vertex) on  vertices has a Hamiltonian cycle if every vertex has degree at least .
Ore: A graph on  vertices with the property that any two non-adjacent vertices have degrees summing to  has a Hamiltonian cycle.
Bondy-Chvatal: A graph has a Hamiltonian cycle if and only if its closure has a Hamiltonian cycle.

Note: A simple graph is a graph such that any two distinct vertices have at most one edge between them, and there are no loops (edges that start and end at the same vertex).


Stars and Bars Theorem

The number of ways to place  indistinguishable balls into  labelled urns is


Note: We represent the  balls by  adjacent stars and consider inserting  bars in between stars to separate the bars into  groups


Pascal's triangle

Pascal's triangle is a triangular array constructed by summing adjacent elements in preceding rows. Pascal's triangle contains the values of the binomial coefficient. 


each  row in Pascal's triangle contains  elements.

Let  be the  element in the  row of Pascal's triangle, with . Then, 
The  row of Pascal's triangle contains the coefficients of the expanded polynomial .
The sum of the elements in the n-th row of Pascal's triangle is equal to .
This is a way to express the identity


Geometry

Triangle:

right triangle properties:

sides of triangle -> angles 

3x,4x,5x
x,x√3,2x -> 30,60,90
x,x,2x -> 45,45,90

 The area of a n equilateral triangle = s² x (√3/4)

 The area  of a square = d² x (1/2)  or s²

 Identify Similar Triangles

 SSS (side-side-side), 
SAS (side-angle-side), and 
AAA (angle-angle-angle), to prove that two triangles are similar

Division of Line Segment
Now, by construction, the triangles PQS and PRT are similar; hence, 

PS/PT = QS/RT = PQ/PR 
http://www.math-only-math.com/division-of-line-segment.html
This is for personal study. Details explanation are available in brilliant.org with examples and practice problems.

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