You’re very right.

2D: If you draw a perhaps wobbly circle shape (loop) on the ground, it has an inside that you can colour in. If your loop is elastic, it can contract to be all in a tiny heap. Topologists call this “simply connected”.

3D: The water on your bath is also simply connected. Your elastic loop, whatever its shape, can shrink back down to tiny.

2D: The surface of your tennis ball is simply connected because any elastic loop on its surface can shrink to nothing, but the surface of your ring donut isn’t, because you could cut your elastic and wrap it arround the donut and it couldn’t shrink because the donut would stop it. Ants living on the surface of the donut might not immediately realise it wasn’t simply connected because they’d never drawn a big enough loop to find out that it couldn’t be shrunk.

3D: The solid donut is also not simply connected, because the ring could contain an elastic band that goes all the way around the ring and back to the start, and it couldn’t shrink to nothing because it would have to leave the donut.

2-Manifolds: a 2-manifold is some kind of surface that doesn’t have an edge and when you look up close it looks like it’s flat-ish. You could make it by sticking lots of tiny sheets of rubber flat to each other but there’s not allowed to be an edge. The simplest 2-manifolds are an infinite plane, the surface of a ball and the surface of a donut. The small ones are called closed. The technical reason for that is to do with not having any edges but still being finite, but you can think of closed to mean finite.

Manifolds may not be as the srrm: If you live in a 2-manifold you might not immediately realise that it’s ball surface and you might not realise it’s a donut surface. If you have a computer game from yesteryear where when you go off the top of the screen you come back on at the same angle and position on the bottom of the screen, and the same for left and right, that’s actually got the same layout as the surface of a donut. To help you see that, imagine your screen was triple widescreen and made of rubber. Roll it up to glue the top to the bottom and then glue the two ends of the tube to each other. You haven’t changed the game play at all but now you can see it’s the surface of a donut shape.

3-manifolds: anything that looks like 3D space up close is a 3-manifold. The simplest 3-manifolds are an ordinary infinite 3D space, a 3-sphere, which is like the 3D version of the surface of a ball, but it’s hard to imagine the 4D ball it’s wrapped around, and the 3D version of the computer game.

The universe: It looks simply connected, but we can’t see that directly, because maybe there’s a very long loop we haven’t gone on yet that gets back where you started without being shrinkable. This is hard to imagine, but it could be like being in the 3D version of the computer game where there’s a long loop that can’t shrink because it goes through one side of the screen and comes out the other before coming back. It can’t be shrink at all, especially not to nothing. The universe is a 3-manifold.

The Poincare conjecture says that every simply connected “closed” (finite) 3-manifold is essentially the same as the 3-sphere. If ALL your loops shrink, no matter how big, and the universe is finite and has no end wall, then it’s the 3 sphere.

Mathematicians have been trying to prove that it’s true for a long long time, and there was a 1M USD prize for proving it that this guy turned down. The prize was largely unnecessary because lots of mathematicians were trying to prove it anyway because it’s so famous and enticing.

And 3 to 25 bunches is between 30 and 250 apples gone bad, so that A few bad apples could spoill half of the apples, and there you have the problem with the police force, especially of the USA where officially it’s not a crime to drive a car whilst having dark skin tone but you can definitely still be summarily executed for it.

That’s path connectedness. Convex shapes are ones in which any two points can be joined by a straight line internal to the shape.

A crescent isn’t a convex shape though, and ironically you left out the word ‘straight’ in your rule about convex.

It’s everything to do with it. Read some of the other replies.

Egregious. I feel your pain.

You’re right on all three counts. It’s not always true, f(x0) has to be L, and the function has to be continuous.

There’s notation for that - (x0 - δ, x0 + δ), so you could say
f(x0 - δ, x0 + δ) ⊂ (L - ε, L + ε)

I don’t think you can use the x0 plus minus delta in the bracket (or anywhere), because then the function that’s 1 on the rationals and 0 on the irrationals is continuous, because no matter what positive number epsilon is, you can pick delta=7 and x0 plus minus delta is exactly as rational as x0 is so the distance to L is zero, so under epsilon.

You have to say that
whenever |x - 0x|<delta,
|f(x) - L|<epsilon.

But I think this is one of my favourite memes.

It’s just that if I were the FBI, or the CIA, or a large criminal organisation, why wouldn’t I be putting a lot of money and the best people I could find on sneaking backdoors for tor into the onion somehow. What a treasure trove of the most potent information there is there! If you can crack tor, you own the keys to the underworld and enough blackmail fodder to get you almost anything you want.

go back to fucking reddit

Yeah, fuck reddit, and fuck u/spez.

Er, am I doing this right?

Why wouldn’t tor be compromised?

Nice.

Rhowch, cwtch, mwyn have to be Welsh. Classicly Welsh sounding words, and mbrsrtowcs, strxfrm can’t possibly be Welsh. Source: my welsh uncle taught me to pronounce Welsh place names.

Wcstold, wcsoll wmffre could be either but sound really weird as Welsh to me.

Thanks for posting the picture of the actual puzzle, it’s what got me playing.

Thank you. Where can I find the wiki?

Edit: Wired says

DuckDuckGo Created a Privacy Exception for Microsoft Cybersecurity and privacy researcher Zach Edwards discovered a glaring hole in the privacy protections of DuckDuckGo’s purportedly privacy-focused browser: By examining the browser’s data flows on Facebook-owned website Workplace.com, Edwards found that the site’s Microsoft-placed tracking scripts continued to communicate back to Microsoft-owned domains like Bing and LinkedIn. DuckDuckGo CEO Gabriel Weinberg responded to Edwards on Twitter, admitting that “our search syndication agreement prevents us from stopping Microsoft-owned scripts from loading”—essentially admitting that a partnership deal DuckDuckGo struck with Microsoft includes creating a carveout that lets Microsoft track users of its browsers. Weinberg added that DuckDuckGo is “working to change that.” (A company spokesperson reiterated in an email to WIRED Weinberg’s assertion that none of this applies to DuckDuckGo search, adding that both its search and its browser offer more privacy protections than the competition.) In the meantime, the revelation blew a glaring hole of its own in the company’s reputation as a rare privacy-preserving tech firm. Turns out this surveillance capitalism thing is pretty hard to escape.

If you’re looking for a never true anticedent reason that “some content and ads you see may not be as relevant to you” is vacuous, that would work if they had an ad browser that was 100% effective on the site in question.

If you’re looking for a never true anticedent for “If trackers are disabled, some content and ads you see may not be as relevant to you.”, it’s that you can’t disable all trackers with a cookie dialog because of the “necessary cookies” blanket exemption, the too many tick boxes to use “legitimate interest” loophole, and that most websites use “fingerprinting”, meaning they reference you not by your cookies but by the worryingly extensive information they get automatically about your browser’s version, settings, capabilities and features, and of course IP address. So it’s never true that trackers are never disabled.

What the Wikipedia article doesn’t explain well in my view, is that logically, “if A then B” means “B or not A” for short, or more explicitly, “in all circumstances, at least one of B, or (not A) , is true”. This is vacuously (emptily) true if B is always true or A is always false, because it’s not genuinely conditional at all.

So I suspect that they meant it was vacuous, not on the grounds that the anticedent could never be true, but that the consequent could never be false. Like “If you give me $10, the sun will rise tomorrow”. In this case, all they need to assert is that “some content and ads you see may not be as relevant to you” is true irrespective of whether trackers are disabled, which is almost certainly what they meant.

I’m curious that the Wikipedia article says the base case in an induction is often vacuously true, but I think they mean trivially true, like cos(1x) + sin(1x) = (cos x + sin x)^1, not vacuously true. I couldn’t think of any induction proofs where the base case was literally vacuous except false ones used for teaching purposes, probably because I could only think of induction proofs of absolute rather than conditional ones. Probably there are mathematical fields where induction is used for conditional statements a lot that I’m forgetting.

You have a total of four choices:

1a. Wipe all their cookies every time, reject them every time they ask.
1b. Wipe all their cookies every time, accept them every time they ask. 2a. Don’t wipe cookies, keep the “essential” ones. 2b. Don’t wipe cookies, accept all our most of them.

2b is the only scenario where you might not get asked again. 1b is the easiest no thanks.

I use the duck duck go browser because it makes that the default and offers to whitelist sites for cookies if you log into them (but you can turn that off in settings). It also autorejects a lot of cookies that use common popups.

Username checks out.