Genady

PS. @Boltzmannbrain, since you seem to like infinities, you might find it interesting.

OK then. Consider the other challenge: prove that it is impossible. IOW, prove that any shape with area <1 can be placed on the plane without touching the grid.

Hint: if you think to take the cube of this expression, you are on the right track.

This is a short-term future, isn't it?

However, as of today, there is no intelligence in "artificial intelligence". It is marketing gimmick. See for example, There’s no such thing as Artificial Intelligence – Australian Data Science Education Institute (adsei.org). Some quotes:

I wouldn't say in this case, "as n goes to infinity", because n doesn't "go" at all here, but rather is one arbitrary member of the set / sequence. Also, the expression, "for all n element of N", doesn't make sense to me in this case. One could write that there is mapping from the set N to set of rows such that every n in N is mapped to a row {1,2,3, ..., n}. See, e.g., Mapping | mathematics | Britannica.

If we will continue the discussion of my proof that the set S is not in the list, the following clarification of the list might help: l=1 L={1} l=2 L={1, 2} l=3 L={1, 2, 3} l=4 L={1, 2, 3, 4} ... Or, equivalently: l=1, L = {x| x∈N & x≥1 & x≤1} l=2, L = {x| x∈N & x≥1 & x≤2} l=3, L = {x| x∈N & x≥1 & x≤3} l=4, L = {x| x∈N & x≥1 & x≤4} ...

The last two lines above are wrong. The set on the line 5 in your list is {1, 2, 3, 4, 5}. This is the set L for l=5, i.e., L = {1, 2, 3, 4, 5}.

L does not have to be in S, and it not in S. L is a set on a line in your list. Each line in your list has a set. L is one of these sets, the one on the line number l.

No, l is not a sequence. It is a number. It is a number of some line in your list. We don't know which line it is, thus we call it l.

L is not a line in S. L is a set on a line number l in your list. If this is clear, I'll go to the second question.

It is not 1, it is l (the letter): L = {x| x∈N & x≥1 & x≤l}

I don't know what you mean in the second statement, but here is a proof that this set is not there. Let's define the set: S = {x| x∈N & x≥1} Let's assume that the set S is in the list. Then there is a line, l, in the list with this set on it. The set on line l is: L = {x| x∈N & x≥1 & x≤l} But L ≠ S which contradicts the assumption. Thus, S in not in the list.

This is wrong, i.e., not every possible set of increasing natural numbers (that increase by 1 starting from 1) is there. The set {1, 2, 3, 4, ...} is a possible set of increasing natural numbers (that increase by 1 starting from 1), and it is not there.

I got a simple solution, perhaps the one you have in mind:

These facts: my father died on my daughter's birthday, and my son was born on my mother's birthday.

Thanks. I got a semi-mechanical way to generate solutions now. Here is another one, not simpler though:

I got the idea. You freeze the sediment on the bottom. Being lighter than water, the chunks of ice with the sediment will float up by themselves. 🤩

We have 9 numbers, 91/8 each. If we remove any of them, we get 8 equal numbers, which can obviously be arranged into two sets of equal sums. If we add all numbers, we get 9*91/8 = 99/8. If we multiply all numbers, we get (91/8)9 = 99/8. They are equal.

It seems to give 76 equations with 83 unknowns. Should be possible. PS. I assume they ask about a non-trivial solution, i.e., all zeroes.

This - my emphasis - is a narrower subset. You might be right about them, I don't know.

I don't think they forget. Plus, most of these "helpers" didn't act in order to help the "millionaire" to become one. They, mostly, did what they did in order to help themselves and their families.

I agree with your both paragraphs. However, I want to add that in my personal experience, I have met and got to know many non-millionaires who are equally "ruthless, extremely selfish, unsympathetic, greedy, and tight... behind a cloak showing a face of care and compassion." I suspect that both categories are just people.

"self made" = "by one's own actions". It says nothing about the environment in which actions are taken. Particularly, it does not say that are taken in empty space / on a deserted island.

Ah, sure. But I don't think this is what a "self made millionaire" means.