Proof by common sense vs induction [Full disclosure: I'm a noob]
The game: Squares and Circles. There are a finite number ($n$) of squares and circles and two players take turns, able to do the following moves:
a) Replace a pair of identical shapes with a square
b) Replace a pair of different shapes with a circle. 
The game ends when there is one shape left. Prove the game ends. 
My questions is whether it's necessary to use induction here to prove that the game ends. Is it sloppy just to consider the two types of moves and show that both the moves decrease the number of shapes by 1, and that after $n-1$ turns there would be only one shape left?
 A: That is a fine proof.  Generally what is an acceptable proof depends on your audience.  You are trying to convince them that it could be made formal if you wanted to.  Making this formal would invoke the fact that the naturals have a minimum, but that is "obvious".
A: Let's see if we can do it in the three ways mentioned in my comment.


*

*One of the three is already done in the posted question; that is the second bullet point in my comment under the question.

*Now let's try the method in the first bullet point in my comment. Suppose there is a smallest number $n$ such that in some cases if there are $n$ shapes, the game need not end. The first move reduces the number of shapes. The number is then less than $n.$ Since $n$ is the smallest possible counterexample and this is smaller than $n,$ this is not a counterexample; therefore the game must subsequently come to an end. Our assumption that a counterexample exists is therefore seen to be false.

*Now mathematical induction. The basic case is $n=1;$ in that case the game has already ended. Now suppose $n$ is some number for which, if the total number of shapes is $\le n,$ then the game must end. If there are $n+1$ shapes, then the first move reduces the number of shapes to one for which the game must end. Therefore with $\le n+1$ shapes, the game must end.


Earlier I posted this related question. The present question seems like a case where the method in my second bullet point is the most efficient way to do the problem. Proving that Euclid's algorithm always terminates is perhaps another such instance.
