A Big Cube and a Little Cube

On his desk, Bengt has two cubes. The small cube has the side 2 dm, and the big cube has the side 1 dm. The reason why the 1 dm cube is "big" is that Bengt claims it has more than three dimensions. Now Bengt wants to place the small cube inside the little cube. (The thickness of the walls of the cubes are negligible, so that for example if you had two identical cubes you could place one inside the other.)

1. How many dimensions must the big cube have for the little cube to fit inside it?
   
2. What if they were spheres instead of cubes?
   

A Slowly Growing Number Sequence

Bengt has found a number sequence. It starts like this:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, 12, 4, 6, 2, 10, 2, 6, 10, 2, 10, 2, 6, 18, 4, 2, 4, 6, 6, 8, 6, 6, 22, 2, 10, 8, 10, 6, 6, 8, 12, 4, 6, 6, 2, 6, 12, 10, 18, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 34, 6, 6, 8, 18, 10, 14, 4, 2, 4, 6, 8, 4, 2, 6, 12, 10, 2, 4, 2, 4, 6, 12, 12, 8, 12, 6, 4, 6, 8, 4, 8, 4, 14, 4, 6, 2, 4, 6, 2, 6, 10, 20, 6, 4, 2, 24, 4, 2, 10, 12, 2, 10, 8, 6, 6, 6, 18, 6, 4, 2, 12, 10, 12, 8, 16, 14, 6, 4, 2, 4, 2, 10, 12, 6, 6, 18, 2, 16, 2, 22, 6, 8, 6, 4, 2, 4, 8, 6, 10, 2, 10, 14, 10, 6, 12, 2, 4, 2, 10, 12, 2, 16, 2, 6, 4, 2, 10, 8, 18, 24, 4, 6, 8, 16, 2, 4, 8, 16, 2, 4, 8, 6, 6, 4, 12, 2, 22, 6, 2, 6, 4, 6, 14, 6, 4, 2, 6, 4, 6, 12, 6, 6, 14, 4, 6, 12, 8, 6, 4, 26, 18, 10, 8, 4, 6, 2, 6, 22, 12, 2, 16, 8, 4, 12, 14, 10, 2, 4, 8, 6, 6, 4, 2, 4, 6, 8, 4, 2, 6, 10, 2, 10, 8, 4, 14, 10, 12, 2, 6, 4, 2, 16, 14, 4, 6, 8, 6, 4, 18, 8, 10, 6, 6, 8, 10, 12, 14, 4, 6, 6, 2, 28, 2, 10, 8, 4, 14, 4, 8, 12, 6, 12, 4, 6, 20, 10, 2, 16, 26, 4, 2, 12, 6, 4, 12, 6, 8, 4, 8, 22, 2, 4, 2, 12, 28, 2, 6, 6, 6, 4, 6, 2, 12, 4, 12, 2, 10, 2, 16, 2, 16, 6, 20, 16, 8, 4, 2, 4, 2, 22, 8, 12, 6, 10, 2, 4, 6, 2, 6, 10, 2, 12, 10, 2, 10, 14, 6, 4, 6, 8, 6, 6, 16, 12, 2, 4, 14, 6, 4, 8, 10, 8, 6, 6, 22, 6, 2, 10, 14, 4, 6, 18, 2, 10, 14, 4, 2, 10, 14, 4, 8, 18, 4, 6, 2, 4, 6, 2, 12, 4, 20, 22, 12, 2, 4, 6, 6, 2, 6, 22, 2, 6, 16, 6, 12, 2, 6, 12, 16, 2, 4, 6, 14, 4, 2, 18, 24, 10, 6, 2, 10, 2, 10, 2, 10, 6, 2, 10, 2, 10, 6, 8, 30, 10, 2, 10, 8, 6, 10, 18, 6, 12, 12, 2, 18, 6, 4, 6, 6, 18, 2, 10, 14, 6, 4, 2, 4, 24, 2, 12, 6, 16, 8, 6, 6, 18, 16, 2, 4, 6, 2, 6, 6, 10, 6, 12, 12, 18, 2, 6, 4, 18, 8, 24, 4, 2, 4, 6, 2, 12, 4, 14, 30, 10, 6, 12, 14, 6, 10, 12, 2, 4, 6, 8, 6, 10, 2, 4, 14, 6, 6, 4, 6, 2, 10, 2, 16, 12, 8, 18, 4, 6, 12, 2, 6, 6, 6, 28, 6, 14, 4, 8, 10, 8, 12, 18, 4, 2, 4, 24, 12, 6, 2, 16, 6, 6, 14, 10, 14, 4, 30, 6, 6, 6, 8, 6, 4, 2, 12, 6, 4, 2, 6, 22, 6, 2, 4, 18, 2, 4, 12, 2, 6, 4, 26, 6, 6, 4, 8, 10, 32, 16, 2, 6, 4, 2, 4, 2, 10, 14, 6, 4, 8, 10, 6, 20, 4, 2, 6, 30, 4, 8, 10, 6, 6, 8, 6, 12, 4, 6, 2, 6, 4, 6, 2, 10, 2, 16, 6, 20, 4, 12, 14, 28, 6, 20, 4, 18, 8, 6, 4, 6, 14, 6, 6, 10, 2, 10, 12, 8, 10, 2, 10, 8, 12, 10, 24, 2, 4, 8, 6, 4, 8, 18, 10, 6, 6, 2, 6, 10, 12, 2, 10, 6, 6, 6, 8, 6, 10, 6, 2, 6, 6, 6, 10, 8, 24, 6, 22, 2, 18, 4, 8, 10, 30, 8, 18, 4, 2, 10, 6, 2, 6, 4, 18, 8, 12, 18, 16, 6, 2, 12, 6, 10, 2, 10, 2, 6, 10, 14, 4, 24, 2, 16, 2, 10, 2, 10, 20, 4, 2, 4, 8, 16, 6, 6, 2, 12, 16, 8, 4, 6, 30, 2, 10, 2, 6, 4, 6, 6, 8, 6, 4, 12, 6, 8, 12, 4, 14, 12, 10, 24, 6, 12, 6, 2, 22, 8, 18, 10, 6, 14, 4, 2, 6, 10, 8, 6, 4, 6, 30, 14, 10, 2, 12, 10, 2, 16, 2, 18, 24, 18, 6, 16, 18, 6, 2, 18, 4, 6, 2, 10, 8, 10, 6, 6, 8, 4, 6, 2, 10, 2, 12, 4, 6, 6, 2, 12, 4, 14, 18, 4, 6, 20, 4, 8, 6, 4, 8, 4, 14, 6, 4, 14, 12, 4, 2, 30, 4, 24, 6, 6, 12, 12, 14, 6, 4, 2, 4, 18, 6, 12, 8...

1. What is this number sequence?
   
2. Will it contain arbitrarily large numbers? Prove it.
   
3. Of the 1000 numbers shown here, the biggest number is 34, and it appears quite early. Is there any particular reason for that?
   
4. Why does the sequence 6, 6, 6 appear in so many places? Does it have something to do with the devil?
   
5. Will any other number ever appear three times in a row? Prove it.
   
6. If you have plenty of spare time: Find the next number in the sequence.
   
7. (hard) Near the beginning, there are a lot of twos. Will they eventually run out, or is there an infinite number of them?