1 Consider the Hilbert hotel and explain how you could fit t

1. Consider the Hilbert hotel and explain how you could fit the following into a Hilbert hotel with no vacancy:

a. 5 new guests

b. Guests coming from a Hilbert Bus with N many seats (that is, a bus with an infinite number of seats, numbered 1, 2, 3, …). Assume that all the seats on the bus are occupied by guests.

c. Guests coming from 7 Hilbert buses with N many seats in each. Assume that all the seats on all the buses are occupied by guests.

d. Guests coming from infinitely many buses numbered B1, B2, B3,…, with N many seats in each. Assume that all the seats on all the buses are occupied by guests.

Solution

a) Since the hotels are infinite, keep moving the guest in nth hotel to n+5th hotel. So first 5 will be vacant for 5 new guests.

b) Move guest in nth hotel to 2n th hotel. so all the odd numbered hotels will be vacant. We can now fit N(countable) guests.

c) Move guest in nth hotel to 8*n th hotel. So we can now fit guests coming from 7 buses as

pth Guest coming from ith bus will take hotel

8p + i .

Since we have vacated all the hotels which are not multiple of 8. we can now accomodate the coming guests.

d)vacate all the odd hotels by moving guest in nth hotel to 2*n th and now accomodate new guests as follows

mth guest from pth bus will take the hotel f(p)^m hotel where f(p) is the pth odd prime