Ceva's theorem

Simple yet powerful theorem that gives the properties of concurrent lines from vertices in a triangle is Ceva's theorem. If the lines AD, BE and CF are concurrent, then  AF/FB x BD/CD x CE/EA = 1. This is attributed to Italina mathematician Giovanni Ceva.

The lines that start from vertices to opposite edges are called cevians. Some examples are medians, angle bisectors, altitudes. Theorem can be used to easily determine whether three such lines are concurrent. Intersection points of such lines are significant centers of triangle in geometry. Some example centers of triangle are centroid, incenter, circumcenter and orthocenter.

References
* Ceva's theorem
* Ceva's theorem
* Triangle Centers

Burnside Lemma

Group theory is formal study for analyzing systems and processes that have symmetry and structure. When an operation is performed on some element, it preserves some structure. Many surprising patterns and discoveries can be made about such symmetries and structures.

Groups are applicable in fields like polynomial equations, polymer structures, topology, number theory, probability, quantum physics and combinatorics. Cauchy, Galois, Cayley, Frobenius, Polya, de Brujin, Redfield and and other several mathematicians contributed to field, but Burnside collected all research and publicized it in his book Theory of groups of finite order in addition to his own contributions to the field.

Group is mathematical abstraction that consists of a set of elements and an operation that satisfies certain properties on the given set. Set of integers with addition form a group. The theory will be more beautiful in dealing with sets of finite order and their symmetry. Set of rotations of rubic cube with combining operation form a group. Symmetry group or permutation group is group whose set is set of transformations like rotation, reflection and moving of an object and whose operation is composition of transformations.

Burnside lemma is a result in group theory that says the number of orbits of input set operated under a group of transformations is equal to the average number of points fixed by the transformations of that symmetry group. It is used to count distinct possible objects or configurations considering the symmetry of objects.

Each possible configuration is an input point. Orbit of a point is set of all possible points possible by applying a transformation of group. Orbits are subset of all points. All elements of orbit share the orbit. Orbits do not overlap and they partition the set of input configurations. Fixed points of a transformation are unchanged points after applying it. Many points may be identical and are simply few transformation away from other points and are so become one orbit. The number of orbits are the number of distinct possible objects or configurations.

Example: Counting the number of bracelets with total four beads of two colors. Two bracelets are considered same if they look identical after some rotation or flipping. There are eight symmetries.

• No movement will keep all sixteen bracelets intact.
• Two bracelets do not change after rotation by one bead.
• Four of them do not change to original after rotation by two beads.
• Two of them do not change to original after rotation by three beads.
• Four of them do not change after horizontal flip.
• Four of them do not change after vertical flip.
• Eight of them do not change after clockwise diagonal shift.
• Eight of them do not change after anticlockwise diagonal shift.
• Total fixed bracelets is  48, that is 16 + 2 + 4 + 2 + 4 + 4 + 8 + 8.
• Total distinct number of bracelets is 48/8 or six.

Group Theory

Burnside Lemma

Burnsides Lemma

Euler

One of the great geniuses ever lived is Leonhard Euler. He contributed to number theory, topology, calculus, infinite series and many branches of the mathematics in 18th century. He started his education in Basel and worked in St. Petersburg and Berlin for most of his life.

Some of the discoveries by Euler

• e^ix = cos(x) + i sin(x) and e^i(pi) + 1 = 0
• v - e + f = 2 to describe relation between verities, edges and faces of polyhydra. 
• Euler line on which orthocenter, circumcenter and centroid are collinear for any triangle.
• Euler totient function to count the numbers that are relatively prime to a number
• Summation of series like ∑ (1/n²) , ∑ (¹/n_!), ...

Leonhard Euler

Euler's identity 

Euler's characteristic

Euler's formula

Euler's totient function

Euler's line 

Euler's biography

Benford's law

Benford's law, also referred as first digit law, is a principle that in any large random real-life sets of numerical data, around 30% numbers begin with 1, around 18 percent begin wit 2, and so on, very small data begins with leading digit 9.  It is a frequency distribution law of nature.

Simon Newcomb, american astronomer, noticed in 1881 that first few pages of logarithm books are used more.  Frank Benford, physicist, paid attention to it in 1938. He collected data from 20 different domains like surface areas of rivers, sizes of populations, physical constants and numbers from a reader's digest magazine.

References
Wikipedia Benford's law
Wolfram Mathworld page

Rule of Seventy Two

"Rule of 72"  is an approximate formula to find how much time it takes for an investment to double with compound interest.  It can also be used in other things that grow similar to interest.

Some scenarios are below. 

• If you invest 100$ with annual percentage of 6%, it takes approximately 72/6 or twelve years to double the original investment. 
• If you invest 100% with annual percentage of 9%, it takes approximately 72/9 or eight years to double the investment value. 
• If the inflation year over year is 4%, the money will lose half of the current value in eighteen years.
• If bacteria grows 3% every hour, it will double in size in twenty four hours or a day.
• As the population of India is growing at 1.2% from 1B people, the population if India will become 2B people in 72/1.2 or sixty years from now.
• If the trend of the GDP growing at 6% annually continues, it will become twice the current GDP in 72/6 or twelve years.
• Suppose an exponential program on n inputs runs for 10 seconds and the increasing input by one takes 11.2 seconds or 12% over 10 seconds, then increasing the number of inputs by six more inputs requires 20 seconds to run the program.

Seventy two is chosen so that it has several divisors and can be convenient for computation. Choosing sixty nine or seventy improves accuracy.  The formula is correct between 6% and 10%, The accuracy for every 3% away from 8% can be improved by adding additional 1 to the value of 72. If you invest 100$ with annual percentage of 25%, it takes 75.66/25 or little more than three years to double the investment value.

Verification from the basic mathematics

S = Sum, P = principal, R = interest rate percentage, r = interest rate fraction = R/100.
The sum after one periodic interval: S = P  + PR/100 =  P(1 + R/100).
The sum after two periodic intervals:  S = P(1+r)(1+r).
The sum after n periodic intervals:  S  = P(1+r)^N

When S = two times the value of P,   2   =  (1+r)^N
Number of periods can be calculated by taking logarithm on both sides.

log(1 + r) = r for small values between 5 to 10%.
N = log(2)/log(1 + r)  = 0.6933/r = 69.33/R

References:
Wikipedia - Rule of 72
Investopedia - Rule of 72
Exponential double time  - Rule of 72
Better Explained - Rule of 72

Ballot problem

In an election where a candidate gets p votes and other candidate gets q votes and p > q, the probability that the first candidate maintains the lead throughout the counting procedure is (p-q)/(p+q).

It was first published by W.A. Whitworth in 1878 and later rediscovered by Joseph Louis Francois Bertrand in 1887.


Sources
Bertrand's ballot theorm
Ballot problem solution