Infinity, take 2, answers.  Note, there are always many ways to do these.  I am giving only one of several possible solutions

1. Suppose the hotel was full, and a bus full of infinitely many people drove up.
A. What would you announce on the loudspeaker?
"Everyone move into the room whose number is 2x what your room number is now."
B. What rooms would the first 3 people on the bus be staying in?
1,3,5

2. Suppose the hotel was full, and 3 busses full of infinitely many people drove up
A. What would you announce on the loudspeaker?
B. What rooms would each of these people be staying in?

bus 1 person 1 room 1	bus 1 person 2  room 7
bus 2 personn 1 room 3	bus 2 person 2  room 9
bus 3 person 1 room 5	bus 3 person 2  room 11

(note, if you work left to right instead of down and up in assigning room numbers, I may mark this wrong, because it does not show that you know a working pattern for how to fit everyone in)

C. Explain your pattern in assigning rooms to people
Take the first person from each of the 3 busses, and then the second person from each of the 3 busses, an then the third person from each of the 3 busses, and so on. Do it so you are always taking people from all of the busses at the same time, and you aren't trying to empty one bus before you start on the next one.

3. Suppose the hotel was empty, and infinitely many busses, each full of infinitely many people showed up. In what order would you assign rooms to people on the bus?

bus 1 person 1
bus 1 person 2
bus 2 person 1
bus 1 person 3
bus 2 person 2
bus 3 person 1
bus 1 person 4
bus 2 person 3
bus 3 person 2
bus 4 person 1

I am doing diagonals, so that each time through, I take the next person on each of the busses I have already taken someone from, and take the first person from the next bus.



4. Show how we know that the set of rational numbers is the same size as the set of counting numbers (i.e. Show a 1-1 correspondence between the set of rational numbers, and the set of counting numbers).

We put the counting numbers on a grid, and come up with a way of assigning counting numbers to them.  Any such counting order has to start somewhere and move out in all directions.  This is one way:


The green line shows the order in which you assign counting numbers.  The dark green numbers show how the first 12 counting numbers are assigned.  The black numbers are the rationals.

5. Consider this infinite set of points.  (for each point the x-coordinate is an even number, and the y-coordinate is an odd number). Suppose that each of these points represents a person, who is going to check into the hotel infinity.  Show how you will fit them all in.  Show enough detail that I understand the pattern.  As part of your explanation, you should at least show who ends up in the first 10 rooms.



My way of assigning counting numbers uses a spiral pattern, starting at (0,-1)