The Locked Boxes Paradox


Imagine an infinite row of boxes, each unlocked, each with a key in its lock.

At time t=\frac{1}{2}, lock Box 1 and put its key into Box 2.

At time t=\frac{3}{4}, lock Box 2 and put its key into Box 3.




At time t=1-\frac{1}{2^n}, lock Box n and put its key into Box n+1.

At time t=1, all the boxes are locked and there are no keys in sight. So the boxes cannot be unlocked!

Can you explain this?

Solution (Click to Show)

Any finite number of boxes locked in this manner can be unlocked. The idea of locking an infinite number of boxes in this way is purely imaginary. This is akin to the philosophical problem raised by Zeno’s Dichotomy Paradox. Therefore, it is not surprising when our expectations that the properties of finite situations will carry over to imaginary infinite situations are not realized.

In other words, it is impossible to describe how the boxes can be unlocked because the process itself is a never-ending one. To unlock the boxes, we would have to get to the last box. But the process is infinite and there is no such “last box”, so there is no box with which to start the unlocking process.

To conclude this post, here’s a short verse which was written by Danish mathematician and poet Piet Hein (1905 – 1996) titled “The Paradox of Life“:

A bit beyond perception’s reach
I sometimes believe I see
That Life is two locked boxes, each
Containing the other’s key.

Posted by

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter` and Facebook. My email is

7 thoughts on “The Locked Boxes Paradox

  1. Mind-boggling! Part of the paradox seems to be that it’s easier for us to imagine an infinite process with a beginning but no end than to imagine an infinite process with an end but no beginning. If we’re allowed the latter, then it actually is possible to unlock all the boxes: at time , take key out of box and use it to unlock box . (This is possible because you’ll already have unlocked box at the earlier time .) It’ll be a blurry beginning, but you’ll have just finished a process with an equally blurry end!

    Liked by 1 person

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