Genady

So? - I was not talking about solving. - I was not talking about "everything." - Math evolves.

You do not need to apologize because I have no idea what you are talking about.

You are applying for a job in a company, and you don't know why?

I also think that because of the aforementioned conceptualization issues, mathematics is the only valid language to describe that.

I think that billions of neurons are just not enough and a brain with trillions or more neurons is required to conceptualize this.

Like in the previous missing area puzzle (https://www.scienceforums.net/topic/131527-find-the-missing-area/?do=findComment&comment=1239116), it is helpful to switch to triangles: The area in question is x+y.

What are advantages of using Latex vs MS Word for math expressions in the posts? At least visually, IMO, the latter is better than the former. Compare Latex: MS Word: Plus, the latter gives more presentation choices such as font, size, etc.

Do they need to stay in the 3x3 configuration?

As we don't have much to do with areas of arbitrary shapes like these, but we know much about areas of triangles, let's make the triangles: The triangles above are named, a to h. The puzzle is, to find the area a+h.

Thank you for asking clarifying questions. There are no gridlines here, only the points, which are the intersection points of the 'imaginary' gridlines. The shape should not intersect any of these points. BTW, to be sure, the OP question has been changed to this: prove that any shape with area <1 can be placed on the plane without intersecting these points.

I certainly agree with you that there is no issue, and I don't try to guess what the fundamental misunderstanding is, but the OP asked to point to an error in his construction, if there is one. I use sets to formalize his construction to point out the error.

It doesn't let me see the whole step-by-step solution, but it looks like it just keeps simplifying the original expression. If so, it misses the beautiful insight: after the first step of taking the cube, the puzzle boils down to the simple equation, which is quickly solved by inspection.

What do you mean, it solves them right away?

So, after taking cube of the expression in question, and simplifying, we get it (i.e., the cube) being equal to Do you see something peculiar here?

No, I don't know.

If we define "a type T set" to be "a set of numbers from 1, increasing by 1, up to a finite number", then there is no infinite set of type T.

Yes, it is. But there is no such set in your construction. IOW, there is no set in your construction "that starts at 1, increases by 1 and has infinite elements."

I understand. There is no set equal to N in your construction. There is nothing to compare to because there is no infinite set in your construction that starts at 1 and increases by 1.

There are never infinite elements and so no such n is needed. There is no such jump.

A.I. says,

It doesn't go through the entire NN every time, but rather a random subset. So, it produces different response every time you run it. I got a similar, correct answer on the 6th trial.

Any comment? Question?

This is a possibility. Or, just hanged around with a black hole for a while.

Yep. This reminded me of my physics teacher who liked to say, "When I ask them any question, they give me any answer."

You are right. +1 Here is my take: