|
I've posed exactly these problems to children about 15 times now. Most children 10 and up I've talked to can figure out the first (one more guest) easily enough, the second (an infinite bus of new guests) with a little help, and the third (infinitely many infinite buses) if I draw a picture. 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... |