Some feature requests to god

A search function to search for things in this world.
A Makefile with target 'clean' so that we can do a 'make clean' to clean up everything (dishes, clothes..everything).

quote from a mathematician!!!

This is a wonderful quote from a calculus text

Logical thinking is much more important than "epsilon" and "delta".

Friendship and the power of pigenhole principle

In a group of n friends (n >= 2), atleast 2 of them will have same no: of
friends.

Reformulating as a graph theory problem
In an undirected graph G (with no self-loops) with n = |V| >= 2, atleast 2
vertices will have same degree.

Proof. We use proof by cases
(case 1). G is connected
The degree of a vertex can be any of n-1 different values 1, 2,...,n-1(A
vertex with degree 0 will imply that G is not connected). There are n
vertices in the graph. So atleast 2 of them should have same degree by
pigeonhole principle.
(case 2). G is not connected
The degree of a vertex can be any of n-1 different values 0,1,...,n-2 (A
vertex with degree n-1 will imply that G is connected). A similar argument
as case 1 applies.

How many digits does 2 "raise to" 1000 have?

We have to find decimal (base 10) digits. So convert to base 10.

2¹⁰⁰⁰ = 10^{(log 2)1000}

Now 10¹, 10², and 10³ has 2, 3, and 4 digits respectively.

So 2¹⁰⁰⁰ should have approximately (log 2)1000 + 1 digits.