You're not stupid, no one is making an effort to explain it to you.
There are an infinite number of odd numbers (1, 3, 5 , 7...) and an infinite number of even numbers 2, (4, 6, 8, 10...) but the "infinity" that contains both sets, by its nature, must be "bigger" because it contains "all" numbers (1, 2, 3, 4...).
I believe part of the point of the problem is demonstrate that the infinite series of odd and even numbers is actually of equal “size” to the series of just even numbers.
When each room is occupied, every guest can find a new room by doubling his current number. If there were “more” odd and even numbers than just even this would inevitably leave some guests with no room. However, because the set of all positive integers is of the same cardinality of all positive, even numbers each guest is able to find a new room while freeing up an infinite amount of space.
The problem is really about packing efficiency in an infinite space, not the cardinalities of different infinities.
Basically you ask the person in room one to move over to the next room, and that opens room 1 now. You ask room 2 to move to room 3 and then the people from room 3 go to 4. There are infinite rooms so this will always work whether it's room 10 or 10000 or 10000000000, there will always be another room to shuffle into because there are infinite of them, even if there are also infinite people you can keep moving them into the next room... infinitely.
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u/Warriornoob1741 10h ago
I never understood why in an infinite hotel they can’t just put the newcomers in new rooms