That's my favorite Rickman scene as well.

It takes a fine actor to pack 30 years of fictional history into a single line. I'll greatly miss his tremendous talents.

It helps a lot to be precise and use concrete examples. "Hilbert has an infinite hotel. We have to be careful with infinities, though, so I'll tell you what that means. If you think of a number zero or bigger, any number at all, Hilbert's hotel has a room with that number on it. Think of a number. ('2 million!') Yep, he's got that room. ('Six googolpillion!') Yep, he's got that room."

"A guest comes into the hotel and asks, 'Mr. Hilbert, do you have one more room?' Mr. Hilbert says, 'Sure!' He picks up his magic phone that calls everyone in the hotel at the same time and gives all the guests exactly the same message. After they do what he says, there's one room empty. What does he tell them to do?"

When they invariably come up with "Move up one room," it helps to belabor a couple of points. First, reformulate it as, "Look at your room number, add 1, and go to that room." (This helps them figure out "Multiply your room number by 2" as the answer to the second problem.) Second, dwell on who goes where, and whether it's a problem. "Where does the guest in room 0 go? ('Room 1.') Doesn't that have someone in it? ('Yes. Oh, no it doesn't, because he went to room 2!')"

No child has figured out my favorite fourth problem, but then it took mathematics until Cantor to figure it out, too.

"An uncountable group of people shows up at the hotel. Let me tell you what that means. They all have infinite name tags, all filled with As and Bs. Every possible name tag is in the group. [Give example names. Blow raspberries to do it.] The head of the group, whose name is 'AAAAAAAAAAAAAA...' [said with a blank look, trailing off] asks Mr. Hilbert if he has room in his hotel. Mr. Hilbert says 'No!' Why does he say that?"

Let them stew for a bit, and ask questions. Going on: "Mr. Hilbert says, 'OK, tell you what. If you give me a room assignment, I can always find someone you left out.' How does Mr. Hilbert do that?"

You can illustrate this with a game, using only four-letter names. Write down something like

AAAA BBAA BABA AABA

"Can you find a four-letter name that's missing?" Play this a few times, and then ask, "Can you come up with an easier way that doesn't make you think of all the names in turn?" Show them how to flip the letters along the diagonal, and then extend to infinite names.

I've had 2 kids and 1 adult follow this to the end. It's always mind-blowing for them, though, no matter how far they get. I follow up with this:

"That stuff they taught you in school, that stuff a lot of people say they hate, is arithmetic. This is math."

EDIT: Come to think of it, I actually helped a friend's 11-year-old daughter decide that she didn't hate math using these problems. She's probably still bummed about being stuck doing arithmetic for now, though...

That's how I'd start. There's probably a way to make the search a lot smarter. Here's the part you're overlooking, though:

"... there are only so [many] formulas made up of a fixed number of terms and operators..."

"So many" = "countably infinite." Paring it down to finitely many would require understanding tantamount to having solved the problem in the first place.

[Edit: I can words.]

But first: another water poloist! Awesome! Water polo was pretty small in my area, so we didn't have a lot of talent in the pool. (Pun intended.) I swam distance on the swim team and was a decent sprinter, so my coach had me play both offensive and defensive hole. My best friend could keep his navel above the water for minutes, so he played keeper behind me. I still love to get my hands on a ball and demo the backwards corner shot. Fun times.

Conscious able swimmers have usually been taught how to lie still and float on their backs, or will figure it out quickly if in calm water. They can often call for help, and will do so, interspersed with periods of rest. (Shouting takes a lot of energy.) If the water is warm, they might have hours to get rescued. If the water is cold or even just lukewarm, how long they have depends on a lot of factors, especially the temperature and whether hypothermia is setting in - but in any case they have a lot less time.

Able swimmers, especially those who are good enough to swim a mile, often underestimate the effects of cold water on the body.

They also tend to underestimate the tremendous strength of moving water. Anybody in moving water who is in distress should be helped ASAP, no matter how good they are at swimming. Moving water is a destroyer and an equalizer.

That leaves us with unconscious swimmers. Nobody swims well while unconscious. :) If there's a chance something could knock you out, whether a boat, an overhanging branch, exposure to cold water, or (for those of extended age) a heart attack, put on a well-fitting life jacket.

In my team the hole would go defend number 3 usually. Defending the other hole is intense.

I see, moving water, cold water and being knocked out. Thanks for the explanation!

When an adult or adolescent drowns, it looks exactly the same, they're much more dangerous to would-be helpers, and half the time they're drunk and surrounded by drunk people.

Head erect, more splashing, perhaps a cry for help, and above all, someone who thinks and acts like a human being. You can't tell from the site, but a drowning swimmer is a wild animal. If you have to approach at all, approach with extreme caution and with something to extend your reach.

It may be you aren't surprised because the premise of the site is to identify people who are drowning. You know it'll be happening within a minute or so. In real life, it'll happen with small probability in the next few hours. It's a lot harder to spot under those circumstances.

Oh, I don't claim that spotting a real drowning in real life will be easy, or even in the video: lots of attention and searching is needed and the lifeguard there does an impressive job. I was only wondering about the drowning behaviour itself.

Watch the face, arms and legs, and water surface. A swimmer in distress has his mouth as high as possible, may be flailing a little with his arms, barely kicks (if at all), barely disturbs the surface of the water, and goes under repeatedly.

But as a lifeguard, I helped weak swimmers, and then requested very firmly that they stay in shallow water. I also took their names so I could get their attention quickly if they wandered too close to the deep end.

I was a lifeguard, so I can speak with some authority. Both skills are necessary. Not only do you have to count and track swimmers, but you have to keep mental notes about their abilities, be aware of what they're doing, and imagine possible outcomes so you can identify and react quickly to the actual outcomes. If your attention wanders for too long, especially at a busy place, you can easily lose track of someone. If you lose track of the wrong person, well, buckle up.

Fortunately, there are usually multiple lifeguards on duty with you, you're usually on a strict 10- or 15-minute rotation with them, and one of the stations is the break room. That helps a lot.

Lifeguarding at a camp is at least 5 times more difficult than at a pool, and I suspect that's why they ran drills to keep the lifeguards on their toes. Underwater visibility is usually next to nothing. The lake is usually full of teenage boys. Access to the water is effectively unlimited. There are often a lot of occluders, such as boats and bushy shores.

I remember this well: sitting in a chair, scanning the pool, and every 20 seconds yelling, "Walk, please!" I was a living reminder to slow the heck down.

I'm pretty sure it was one of my most important tasks, though. The only serious injury I ever witnessed (a skull fracture) happened when someone was running on deck.