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A Makefile with target 'clean' so that we can do a 'make clean' to clean up everything (dishes, clothes..everything).

## Monday, October 27, 2008

## Thursday, October 2, 2008

### quote from a mathematician!!!

This is a wonderful quote from a calculus text

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

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

Posted by
Balagopal

### 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.

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.

Posted by
Balagopal

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

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

2^{1000} = 10^{(log 2)1000}

Now 10^{1}, 10^{2}, and 10^{3} has 2, 3, and 4 digits respectively.

So 2^{1000} should have approximately (log 2)1000 + 1 digits.

Posted by
Balagopal

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