Is Contra-positive and Converse statements just different way of saying a if then statement? So i had this question :-

Write, If a natural number is odd, then its square is odd in different ways.

I had included these statements in my answer :-

$(1): -$ If the square of a natural number is odd then the number is odd
$(2) :-$ If the square of a number is not odd then the number is also not odd

But to my surprise none these were in the answer key.
Answer key :-

$(1) :- $ A natural number is odd implies that its square is odd
$(2) :- $ A natural number is odd only if its square is odd
$(3) :- $ for a square to be odd it is sufficient that the number is odd

So my question is,
Are Contra-positive and Converse of a statement, different ways of writing a statement or not ?
I am a bit confuse as they mean the same thing as the original sentence.
 A: The contra-positive is an equivalent statement to an implication, while the converse is generally not.
For example, take the true statement "if $x=1$ then $x^2=1$". The contrapositive is "If $x^2\neq 1$, then $x\neq 1$", which is a true statement, while the converse, "if $x^2=1$ then $x=1$" is false ($-1$ is an $x$ that satisfies the "if" part)! 
A: The converse is only true when you have an "if and only if" statement.  The contrapositve is always true.
Suppose I work every Monday and alternate Tuesdays.
I can make the statement:  "If it is monday I am work today."
The Converse:  "If I am at work, today is a Monday." is not necessarily true.  It could be a Tuesday. 
The Contrpositive:  "I am not working today, therefore it is not Monday."  Is true.
In your example $x^2$ is odd $\iff x$ is odd.  The arrow points both ways (if and only if).  The converse is true when the statement is true, and the statement is true when the converse is true.
If you are asked to prove and if and only if statement, you must prove both directions.
A: All three statements in the answer key are restatements of the original. They are different ways of writing the same thing.
The original has the form "if X, then Y".


*

*Answer (1) has the form "X implies Y".

*Answer (2) has the form "X only if Y".

*Answer (3) has the form "X is sufficient for Y".
Note that these answers are neither in converse form nor in contrapositive form.


*

*Converse of original: "If Y, then X" (not logically equivalent to original)

*Contrapositive of original: "If not Y, then not X" (logically equivalent to original)
A: In your question, the statement and its converse happen to both be true, but I wouldn't say that they are "the same" or that they "mean the same thing". 
The statements "$1+1=2$" and "there are infinitely many prime numbers" are also both true. But are they just two different ways of writing the same statement? 
Of course, what is meant by "the same" depends on context, but most of the time one wouldn't want to view these statements as the same. One reason for this is that while they can be proven to be logically equivalent (in the trivial sense that they're both true), such a proof requires you to know what the words involved mean (like natural number and odd number and square), i.e. they require some background non-logical assumptions. 
On the other hand, "If a natural number is odd, then it's square is odd" and "A natural number is odd implies that its square is odd" and "If the square of a natural number is not odd, then the natural number is not odd" are all logically equivalent by virtue of their form - you don't need to know the meanings of the non-logical words to see that they are equivalent. 
