Well first, let's see this example.

So, we have initially this big triangle, which is an obtuse triangle.

And you see that it is cut into seven pieces, and those pieces are triangles.

And actually, all of the pieces are acute.

So we have just proven the theorem that states that this is impossible,

that we cannot cut a triangle into acute pieces if the initial triangle is obtuse.

But on the other hand, we have this example, which proves that it is

possible to cut this particular obtuse triangle into acute pieces.

So what was wrong?

The example is obviously correct, so

probably something was wrong with the proof, and it is often the case.

We have to be very careful when making the proofs to

not make some incorrect logic in our proof.

So let's look at what was wrong.

Actually, the induction step assumed that if we cut a triangle into several

triangular pieces, we can do it by several steps of cutting a triangular

piece into more triangular pieces, and actually into two triangles.

So we proved by induction that if we cut our initial obtuse

triangle in a particular way, when every time we take a piece and

cut it by a line into two triangular pieces,

then it is impossible to cut the obtuse triangle into acute triangles.

But this is not the only was to cut the triangle.

We don't have to always cut the triangle into exactly two pieces which

are triangles, we can cut it into more pieces.

And in this previous example that I've shown you, you see that there is

no such thing that we cut our initial obtuse triangle into two pieces.

We cut it into seven pieces at once.

And so this is the case that our proof didn't consider.

And this is where we're wrong.

So the induction step itself is correct and the induction base itself is correct.

But the assumption is incorrect, that the only way we can

cut an obtuse triangle into triangles is to cut one by one.

This was true when we're cutting the plane with lines,

that we can add lines one by one and we can get the same picture on that.

And it was also true for segments connecting the points that we can

add segments one by one, and then we will get the final picture.

But it is not true that, for any way of cutting a triangle into pieces,

into triangular pieces, we can simulate this by cutting first a triangle

into two pieces and one of those two pieces into two, and so on, and so on.

This is just not true because there are different ways to cut the triangle.

And so this is an example where mathematical induction actually fails.

Well, actually this is not the fault of any induction, it is faultless.

It is our fault that we applied it in a wrong way.

And this shows that you have to be very careful.

This is a very powerful method that you have to handle it with care,

not to make a mistake.

And in the next video, I will show you even more flawed

induction proofs which will show you what not to do.

And it is equally important when you have a powerful method in your hands,

not only to understand how to apply it, but

also how to not apply it when it is not applicable.

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