1. Suppose that you are working at the main desk of the Hotel Infinity.  Each and every room is occupied.  Someone comes to the hotel and wants room.  Tell two different ways you could make room for the new person (Everyone in a room at the beginning is in a room at the and, and no one has to share). Tell what room the new person goes in.

Way #1: [Announce over the loudspeaker]  Everyone move to the room whose room number is one higher than the room number you are in now.  New person in room 1.

Way #2: Everyone in an odd number room, move to the room whose number is 2 higher than the room number you are in now.  New person in room 1.

Way #3: Everyone in an even number room, move to the room whose number is 2 higher than the room number you are in now.  New person in room 2.

Way #4: Everyone in a room with a number 10 or greater, move to the room whose number is 1 higher than the room number you are in now.   New person in room 10.

Way #5: Everyone in a room whose number is a multiple of 10, move to the room whose number is 10 higher than the room number you are in now.  New person in room 10.

(lots more ways are possible)

2. Suppose that you are working at the main desk of the Hotel Infinity.  Each and every room is occupied.  Four more people come to the hotel and want rooms.  Tell two different ways you could make room for the new people (Everyone in a room at the beginning is in a room at the and, and no one has to share). Tell what rooms the new people go in.

Way #1: [Announce over the loudspeaker]  Everyone move to the room whose room number is 4 higher than the room number you are in now.  New people in rooms 1, 2, 3, 4.

Way #2: Everyone in an odd number room, move to the room whose number is 8  higher than the room number you are in now.  New people in rooms 1, 3, 5, 7.

(lots more ways are possible)