Genady

I do not.

Yes, I just don't see increasing as similar to expanding. But it is perhaps no more than unimportant semantics. One can, no problem. Friedmann–Lemaître–Robertson–Walker metric - Wikipedia

Took a picture of a resting hummingbird yesterday: Only after enlarging it at home and looking carefully, noticed a lizard resting nearby:

Why is it so?

If you shift the shape up, or the grid down as below, the shape does not intersect with the grid dots:

The expansion / contraction of space is clearly expressed in the Robertson-Walker metric. I am not familiar with a concept of expansion of/in time.

In case someone wants an explanation:

Are we? For what purpose? In what sense? Boundary between what and what? Between past and future, perhaps. But this is a temporal, not a spatial boundary. Every spatial point in the universe was a 'center' of expansion then, as it is now. PS. Remember, that we are talking here about an ideal homogeneous and isotropic universe. "Brooklyn is not expanding."

This style of writing from an intelligent person is painfully familiar to me.

No, of course it is not wrong. Every place in the universe you pick, was inside that hot dense stuff that was the universe content at the time of Big Bang.

Curious. I didn't know it needs explanation and more so, there is a way to explain it.

Is it this book? I have it for some time, but there always are other books in front of it somehow...

Thank you. I understand that xxx * 7xx = xxxxxx but I don't know what other information, if any, it gives me, because the Russian way of long division, which I did in school, was quite different: DIVIDE THE NUMBERS [RUSSIAN STYLE] - WITH REMAINDER explained by Photomath - YouTube (I assume that the x's are just placeholders.)

Sorry, but I still don't get it. Even the case of empty set is not "a complete absence of things", because there has to be a set for it to be empty, and this set is present rather than absent.

I've made a mistake by posting this puzzle in the other thread (https://www.scienceforums.net/topic/131412-puzzle-for-the-day/?do=findComment&comment=1239451). It does not belong there. Thus, I repost it here and copy relevant comments from that thread. Hope to continue discussing this puzzle here: @TheVat: Seemed easy. I'm on a tablet so the hidden button doesn't seem to appear. It can be figured without recourse to formulae. @Genady: It is an easy one to make a mistake, too. @TheVat: @Genady:

It is an easy one to make a mistake, too.

I am not aware of it. What is "a complete absence of things"? I don't understand it. Do you mean, in math? An empty set?

It is straightforward to get the recursive formula from this: X(n+1) = (n+1)^2 + 4(n+1) - 1 = n^2 +2n +1 + 4n +4 - 1 = n^2 + 4n - 1 + 2n + 5 = X(n) + (2n + 5) The formula, X(n+1) = X(n) + (2n+5), makes it easy. The additive term, 2n + 5, gives the consecutive odd numbers starting with 7 (for n=1).

It can be. Take a look at these slides from Alan Guth's lecture in MIT:

Imagine that you got planks 1m long each, as many of them as you wish. The planks have a width and a thickness, but those are not important here. What is important, they have weight. The only thing you can do with the planks is to put them on top of each other, like this: Your goal is to make the tower with the largest possible "shadow", x. IOW, the largest horizontal distance between the leftmost and the rightmost edges of the planks. The question is: what is the largest x so that the tower and its parts don't fall?

Unlike the BH case, necromancy still has a chance 😉