[BetaCygni]

As a non-mathematician, for a 2d circle on a flat plane:

- The edge of a circle is curved (convex)

- It does not matter how small a piece you cut of the edge, there will always remain a convex curve.

- The edge of a square has straight edges, so you cannot put this piece at the edge. This means the old outside edge will have to be on the inside.

- You cannot fit the convex edges to each other or to a straight edge.

- Cutting a concave edge from the inside of the circle to fit the convex edge from the outside to will not help as it will produce a new convex edge.

Ergo: there is no place to put the convex outside edge of the circle, so you cannot turn it into a square.

[pfortuny]

So not so OBVIOUS, anyway. You need the notion of convexity (not so easy to give abstractly unless you already know the definition). Items 4 and 5 look clear but from there to obviousness...

Being visually intuitive is very different to being obvious.

[authoritarian]

This seems a bit pedantic. I'm certainly not a mathematician but it does seem obvious that you cannot cut a circle into any number of pieces and rearrange it to be a square, as a circle has curves which would prevent you from making the square a solid without gaps

[pfortuny]

No no; I’m trying to explain that “obviousness” is a bad idea to prove anything...

Like “an infinite te has an infinite branch”...

[epanchin]

You could create a concave edge with a series of triangles (really, trapezoids).

Ps. I suppose this would really only work for an approximate circle, which is a polytope.