Infinity 145
Yes, indeed. The manager again directs the resident of room n to move to room 2n, which
again frees up the odd-numbered rooms. And then, the manager directs the passenger in
car c, seat s to check into room 3
c
5
s
.
Room n → Room 2n
Car c, Seat s → Room 3
c
5
s
Since 3
c
5
s
is certainly an odd number, it is available for a new guest, and different train pas-
sengers will take up different rooms because of the uniqueness of the prime factorization.
So again everyone is accommodated.
Hilbert’s half marathon
Everyone was in town for the big race, and here it comes: Hilbert’s half marathon—this
crowd likes fractions—with the runners densely packed in amongst themselves. The run-
ners each have a race number, one for every nonnegative rational number. They all want to
check into the hotel.
1
10
3
10
7
10
9
10
11
10
13
10
17
10
19
10
21
10
23
10
27
10
29
10
31
10
33
10
37
10
39
10
41
10
42
10
43
10
1
9
2
9
4
9
5
9
7
9
8
9
10
9
11
9
13
9
14
9
16
9
17
9
19
9
20
9
22
9
23
9
25
9
26
9
28
9
29
9
31
9
32
9
34
9
35
9
37
9
38
9
1
8
3
8
5
8
7
8
9
8
11
8
13
8
15
8
17
8
19
8
21
8
23
8
25
8
27
8
29
8
31
8
1
7
2
7
3
7
4
7
5
7
6
7
8
7
9
7
10
7
11
7
12
7
13
7
15
7
16
7
17
7
18
7
19
7
20
7
22
7
23
7
24
7
25
7
26
7
27
7
29
7
30
7
1
6
5
6
7
6
11
6
13
6
17
6
19
6
23
6
1
5
2
5
3
5
4
5
6
5
7
5
8
5
9
5
11
5
12
5
13
5
14
5
16
5
17
5
18
5
19
5
21
5
22
5
1
4
3
4
5
4
7
4
9
4
11
4
13
4
15
4
17
4
1
3
2
3
4
3
5
3
7
3
8
3
10
3
11
3
13
3
1
2
3
2
5
2
7
2
0
1
2
3
4
Can the hotel manager accommodate all the runners? Yes. After freeing up the odd-
numbered rooms as before, he can place runner p/q in room 3
p
5
q
, where p/q is in lowest
terms.
Room n → Room 2n
Runner
p
q
→ Room 3
p
5
q
In this way, the entire half marathon fits into Hilbert’s hotel. It might be a little surprising
at first how similar the manager’s solution was for the half marathon to the solution for the
train. But ultimately, it should not be too surprising, because every train passenger is
determined by two positive integers—the car number and the seat number—and similarly
with every marathon runner: the numerator and the denominator.