Strange paradoxes of recursive universes

I couldn't believe my own answer to this puzzle. After thinking hard to disprove it, I finally understood it.

Imagine being in a large box with your box lying on the floor beside you. Your box contains a smaller copy of you and a smaller box inside and so on ad infinitum. You reach your hand into your box and pull out the smaller box. As you do this, you see a bigger hand coming from above and taking your box. Note that all the infinite copies of you have done the same. The boxes and the hands can pass through each other. So, there are no collisions.

Then, you put the box in your hand on the floor where the old box was. Where is your box now?

In this picture, your box is colored. The box that you pulled out is shown with a thick border. Dots indicate that the pattern continues forever.

The solution

As insane as this sounds, your box is now infinitely far away.

Let's derive the answer. To keep track, we will assign numbers to the boxes. Your box is 0. The boxes inside will be labeled 1, 2, 3, 4, .... You are inside the box -1. We label the outer boxes -2, -3, -4, .... Here's what it looks like:

Let's use a simple notation to indicate that we can grab box 2 through box 1 (or that 2 is inside 1)

[1] -> [2]

Then, the original configuration of the boxes looks like the following.

... [-2] -> [-1] -> [0] -> [1] -> [2] -> [3] -> [4] -> [5] ...

You pulled box 1 into -1, 0 was pulled into -2, -1 was pulled into -3, -2 was pulled into -4 and so on. We can observe the emergence of two chains - one for odd numbers and one for even numbers.

... [-5] -> [-3] -> [-1] -> [1] -> [3] -> [5] ... (odd numbers)

... [-6] -> [-4] -> [-2] -> [0] -> [2] -> [4] ... (even numbers)

And this is the final result. From your perspective, the even numbered boxes including your box are nowhere to be seen!

We are left with a rather disturbing mystery. Are half of the boxes really gone and when exactly? Can't we just reverse our actions to get them back?

Resolving the mystery

Being a recursive arrangement of boxes within boxes, the connections between boxes were constrained. To reach an inner box, you have to go through the boxes in the middle. So, if one of the boxes in the middle cannot be reached, you lose access to all the boxes inside it - exactly what has happened here. The moment you placed the smaller box on the floor and let it go, you ended up creating two parallel universes. By going inside or outside a box you cannot jump into the other universe.

This concept is often encountered in data structures like linked lists. A linked list is a chain of nodes. Each node contains a number and the address of the next number. If you remember the starting node, you can keep jumping to the next node's address and see all the numbers. But, if you cut the linked list in the middle or take alternate nodes and link them together into 2 linked lists (like above), you cannot reach all nodes starting from any one node.

Simpler example

Sometimes it helps to look at a smaller (finite) example. Imagine there are just 8 boxes.

[-4] -> [-3] -> [-2] -> [-1] -> [0] -> [1] -> [2] -> [3]

The first 2 boxes (-4, -3) are not moved. Both -2 and -3 will be inside -4. -2 has all the even boxes inside and -3 has all the odd boxes inside.

[-4] -> [-2]  -> [0]  -> [2]
  ↓  
[-3] -> [-1] -> [1] -> [3]

Aha! The largest box still connects the odd and even numbered boxes. But, that connection was pushed away to infinity in our puzzle.

This is a modified version of a puzzle created by Michael Savage.