Why the Magic Paving Stones Puzzle is a Paradox

The puzzle posed in Magic Paving Stones may or may not be a paradox.  It might come down to expectations and definitions.  If you don’t have any expectations, or you work out the right answer straight away, then there’s no paradox at all.  If there’s a lack of clarity in definitions of key terms, then … well, it’s still a sort of paradox.  Alternatively, it might be one of Quines’ “veridical” paradoxes.

The crux of the puzzle is how many extra steps you must make a monster take in order that it never reaches you.  If you can ensure that there is an extra paving stone inserted between you and the monster for every step it takes then it can never reach you, so the problem then becomes how many extra paving stones across the whole range need to appear (at random, as defined in the puzzle) every time the monster is about to take a step.

It's not 1, as suggested by one respondent (at r/paradoxes).  If only one step is added, then, as soon as the monster takes one step, there is a (possibly very small) chance that any step that appears (by means of an additional paving stone) will be behind it – on the other side from you, allowing the monster to get closer to you and thus increasing the chance that the next paving stone that appears will also be behind it.  That leads, quite quickly, to a situation in which the monster is bearing down on you steadily and remorselessly like the antagonist in “It Follows”, but you can’t run!

It's not 2 either, as I have heard suggested.  This is, I think, a quite intuitive answer – there is one paving stone to counteract the step that the monster is about to take and another to push it further away.  The problem is that there is a vanishingly small, but non-zero chance that both additional paving stones will appear behind the monster (at some time after it has taken its first step) and it therefore gets closer to you.  Again, this will increase the likelihood, in future rounds, that both additional paving stones will appear behind the monster.  Sure, the time taken for the monster get to you increases, but never is very long time (to paraphrase Roxette and the Red Hot Chili Peppers).

The other intuitive option, which only seems to be suggested when I use an actual number of paving stones for the initial path, is to set the number to equal the number of existing paving stones or number of slots available.  So, say I suggested (as an example) that you start off with 12 paving stones, it seems intuitive to some that the answer is either 11 or 12.  I think there is a tendency to think of the slots being filled evenly, but that was not what was specified in the puzzle.  The likelihood of a paving stone going into any specific slot is equally distributed across all slots, which does not prevent multiple paving stones appearing the one slot.  Once the monster has taken one step towards you, there is a possibility that future magic paving stones will appear behind it and eventually it becomes quite likely that all of them will, allowing monster to get closer to you and eventually get you.

I wrote a program to simulate this, allowing me to adjust the total number of rounds, the initial location of the monster, path length and the preset number of magic paving stones.  As an extreme, I set the monster at paving stone 2 of 2, and then set the number of magic paving stones to 12.  Then I charted the output:

Note that this is isn’t always the precise outcome quantitatively, sometimes the number of rounds was at least eight times that, sometimes as little as half of it.  But in every single instance when I ran the numbers, it turned out that – eventually – the monster gets you.

So, the answer to the puzzle appears to be that you need an infinite number of paving stones to appear each round in order to prevent the monster ever getting you.

I don’t think that this is an intuitive answer, nor the answer that most people will expect.  So, in the soft sense at least, I do think that this is a paradox.

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There’s another sense in which this is a paradox – based on the vagueness of the term “never”.  The question was posed like this: what is the minimum number of magic paving stones do you have to preset for activation each round to ensure that the monster never reaches you?

If I had said “ensure that the monster cannot reach you”, then perhaps not even an infinite number of magic paving stones would suffice.  As soon as we start throwing around infinite numbers, we must sure accept also the possibility of an infinite number of rounds – after which the monster does reach you.  The way I think of it is that either the monster gets you, and the game is over, or there’s another round with an infinitesimal likelihood that that monster will get slightly closer to you.  If there’s no end of rounds, then the only way it stops when the monster gets you.

This seems like a weird variation of the Hilbert paradox, which is a veridical paradox, not because it involves set theory, but because it involves a conflict of infinites.

In the first round, you effectively put an infinite number of paving stones between you and the monster. In the second round, you have an infinite number of paving stones that appear across a range that is infinite, but split between single paving stone behind the monster (that it just stepped away from) and the infinite number between you and the monster (noting that it is a slightly larger infinity, since it is the infinite number of new paving stones plus the original number of paving stones – let’s call this infinity+).  There’s a one over infinity+ likelihood that any one of the infinite number of paving stones will appear behind the monster, but there’s the preset and slightly smaller infinite number of them.  The likelihood of any one paving stone appearing behind the monster is infinitesimally small (against what is effectively a 100% chance of appearing between you and the monster, since infinity/(infinity-1)=0.99999…=1). But with an infinite number of paving stones appearing – all possibilities no matter how unlikely will surely be expressed. 

Now there are two infinities of paving stones, with some tiny proportion of them being behind the monster.  With each round another infinite number of paving stones appears, each with a slightly greater proportion of them appearing behind the monster, but the monster is still getting further away.  It seems that it is impossible that it will ever reach you.  However, if we use any arbitrarily large number, N, as the number of magic paving stones, we can see that – eventually – the monster does get you.  Adding one, or two, or any other arbitrarily large number to N doesn’t save you and there’s no reason to think that that would ever change and, logically, we could add an arbitrarily large number times an arbitrarily large number and it would make no difference, so neither should adding infinity.  And thus the monster gets you, eventually, even though it seems quite impossible that it should.

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So, I ask again, is this a known paradox?  Or a variation of a known paradox?  (Or just silliness when you take it to extremes and get infinities involved?)

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It has been pointed out (by u/crescentpieris) that there is a very similar mathematical puzzle that appears paradoxical, thus falling into the soft paradox category – Ant on a Rubber Rope. I do have one more twist to it though, as per the next article.