Accelerated Twisting on the Magic Paving Stones

In The Unexpected Result of Twisting on the Magic Paving Stones, I showed that the scenario behind the Magic Paving Stones puzzle/paradox – at least when twisted – correlates with the notion of the OE curve, in a FUGE universe.  The monster correlates with a photon, the magic paving stones inflate the path/universe at a rate that is proportional to the preset number, and you are an observer.  While it sounds like being reached by a photon is not quite as dire as being reached by a monster, I never did actually say what the monster would do when it reached you, perhaps all it would do is trigger one of the photoreceptor cells in your eye.

When I mentioned this at reddit, I noted that expansion of the universe is thought to be accelerating in the Standard Model.  But my thinking was that since this is equivalent to an ever-increasing number of magic paving stones being added each round (on either side of the monster, in front and behind), it would make sense that the monster would still get you – eventually.  There would still be a possibility of paving stones appearing behind the monster, meaning that it might take longer but the monster would still get you.

I modified my code to incorporate expansion such that, each round, the pathlength would increase by a truncated percentage.  For example, if the initial pathlength was 25 paving stones and the percentage selected was 4%, then in the first round the number of paving stones would be increased by 25*0.04=1, then they would increase by one each round, for 25 rounds, until there were 50 (50*0.04=2).  Then they would increase by 2 each round for 13 rounds, until there were 76 (76*0.04=3.04), then the increase would be 3 for 8 rounds, 4 four 7 rounds, 5 for 5 rounds, 6 for 4 rounds, 7 for 4 rounds, and so on.

What I found was that, for an increase value of 4% and for initial pathlengths of greater than 25 with the twisted scenario (so the monster is one step closer at the start), there was small chance that the monster would not get you.  That chance increases rapidly as you increase the initial pathlength.

The most common result with an initial pathlength of 26 was something like this:

A far less common result, for which I had to run the simulation 79 times the first time, 383 times the second time and 322 times the third time before it occurred (and the fourth time I exited after 400 refreshes without any triggering), was this:

That little kick up at the end is more extreme (occurs earlier and is more frequent) when the initial pathlength is greater, for example using 40 when it manifested immediately five times in a row:

This is interesting, but we need to be careful about these chunky initial states, by which I mean states in which the random influences are comparatively large. I wanted to look at a less chunky scenario, specifically where the initial pathlength is 801, the monster starts on 800 and the percentage increase per round in 0.125% (=1/800).  My intuition (which was wrong) was that this would also be unstable.  It’s not, I could refresh numerous times and the monster gets you every time.

It only gets unstable (for a percentage of 0.125%) at a pathlength of about 1032-ish at which point monster only gets you roughly half the time – of course it happens earlier but at a lower frequency.

A result like the above is not unusual with those values (although there’s usually either a kick at the end, or the monster has got you before 10,000 rounds).

There seems to be a pattern here, if we increase the initial pathlength and decrease the acceleration proportionately, we get this:

In this case the monster got you, but only after a very long time.  There are so many calculations with values of this size that my machine slows down too much to carry out many refreshes, and increasing the pathlength by another order of magnitude would be an overnight (or weekend) task.

Keen-eyed readers will note that I increased the initial pathlength by very slightly more than a factor of 10.  This is because I was intending to test the notion that the instability kicks in when the pathlength equals the inverse of the percentage increase per round times the square root of one and two thirds.  While it does seem to really manifest at about pathlength=1.29*1/percentage, 1.29 is not the actual value (and neither is 1.29099444) – the monster seems to get you every time with the values above (usually within about 40,000-60,000 rounds).

The precisely nature of this effect, and the mechanism behind it remains unclear.  What is clear though is that the closer the monster is at the beginning, the more likely it is to get you:

A single paving stone is actually enough to massively increase the likelihood that the monster will get you, or in this case about 0.1% closer.

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The question that I have, of course, is whether this in any way correlates with the accelerated expansion of the Standard Model universe during the Dark-Energy-Dominated era.  We’d have to think about the time at which that era commenced, about 4 billion years ago, at which time the Hubble parameter value was about 66.5 km/s/Mpc (because in the Standard Model, in that era, H(t)=2/(3t) where t is the time elapsed since the Big Bang).  Assuming that the Hubble parameter has increased uniformly since then, to reach the current value (H(t)≈1/t≈70), and using a year as our unit of time, this would equate to an increase (Δ) given by:

Δ=^4000000000√(70/66.5)-1=1.28×10^-9%

This is using a 4 billionth root, but of course the granularity of the universe isn’t at the scale of years, it’s about 6×10⁵⁰ times finer, meaning that the increase per Planck time is in the order of 2×10^-60%.  (This is because x=y^1/z-1, where y-1→0, then y^1/(z*w)-1→x/w.  I asked whether there is proof for this at Reddit and was given one.  I don't doubt that proof, but it's one of those back of the napkin things that will work for a mathematician but have shades of appeal to authority for anyone else.  But that is better than the brute method that I had which was to just keep inserting values in and find that it works every time, which you can try yourself if you prefer.)

By implication from the above, it would seem that a photon would reach us reliably (albeit after a long time) if it were emitted a little bit less than about 6×10⁵⁹ Planck lengths away, which is 3.5×10¹⁶ light years, or about 35 million billion light years.

If the logic above is correct, then the claim that a photon that is only 15 billion light years away would never reach us would be incorrect.  Perhaps the nature of the acceleration of expansion in the Standard Model is not as I have described.

The Unexpected Result of Twisting on the Magic Paving Stones

At the end of Twisting on the Magic Paving Stones, I noted that there was something unexpected.  Please look back over previous articles to see the full details but, in short, what I was discussing is a scenario in which, in each round of a sort of game, a number of magic paving stones (≥2) are inserted at random locations (with an evenly distributed likelihood) into a path, then a monster takes a step from one paving stone to the adjacent one closer to you (starting at one step closer to you than the last paving stone in the path) and then you eliminate one paving stone of your choice (and since you don’t want the monster to get you, you are going to eliminate one behind it, further away from you).

Set the number of randomly inserted paving stones to 2, the initial pathlength to a sufficiently large value (for which I choose 120, so the monster starts on paving stone number 119) and plot the output once every 200 rounds and you get this:

It might not be immediately obvious, but we’ve seen something like this before (or at least those of us who looked at the OE curve).  To emphasise this, here is that monster location curve with the OE curve below it – noting that smoothing has been turned off.

This is rather curious and was in fact something that I was struggling with – having previously posted two articles on the same sort of thing before retracting them (soon to be reinstated at Observable Events Curve - Is Double Dipping Essential? and Observable Events Curve - Not Quite a Drunkard's Walk).

Note that the time (or number of rounds) that is taken for the monster to get you (or “capture time”) is not set, there’s something akin to chaos happening in that the capture time very much depends on slightly different conditions early in the scenario, which – after the first round – are randomly imposed.

The OE Curve line in red above is based on time it takes for the separation to reach zero, t₀ (with a relatively insignificant offset to account for the fact that the initial separation is not zero, but rather an offset that we can call x_i).  The final equation becomes:

x'=ct((ct₀+x_i)-ct)/(ct₀+x_i)

If ct₀>>x_i and ct=x, this approximates, of course, x'=x(ct₀-x)/ct₀.

Twisting on the Magic Paving Stones

In Why the Magic Paving Stones Puzzle is a Paradox I provided the solution to the Magic Paving Stones puzzle. 

What I want to do now is to introduce a slight twist.

Once again, you are standing on one end of a path of paving stones.  At the other end of the path is the monster.

Once again, you have the choice again about how many magic paving stones will appear randomly (as previously defined) along the path every round, prior to the monster taking each step from one paving stone to the next towards you.

However, before things start, you are offered the option to give one free step that the monster can take towards you (so before any magic paving stones are activated) in exchange for the ability to destroy one paving stone of your choice every round, so long as the sum of additional paving stones per round is greater than zero.

I know this is complicated, so I will try to clarify using an illustration.

What choices do you make in this instance?  Can you prevent the monster from getting you?

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Note that there’s something unexpected involved here, but it’s not actually in the solution to the puzzle.

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.

Magic Paving Stones

This is a sort of a puzzle, sort a paradox (in the soft sense of running contrary to expectation, rather than involving explicit self-contradiction).

Imagine you are standing the first paving stone of a path that consists of an arbitrary number of paving stones.  At the end of the path is some vague monster that you don’t ever want to reach you.  As happens in a nightmare, your feet are firmly stuck to the paving stone that you are standing on and you can’t run.  Fortunately, you have two facts in your favour.

Fact one: the monster moves at a rate of one paving stone per round.

Fact two: you have a singular (once-off) choice to preset the number of magic paving stones that will be activated per round – just before the monster moves.  These magic paving stones will appear between existing paving stones, at random.  More specifically, the likelihood of the paving stone appearing between any two existing paving stones is evenly distributed across the whole path.  For example, if there are three paving stones, there are two locations where a new paving stone could appear – between #1 and #2 (slot #1#2) and between #2 and #3 (slot #2#3), with equal likelihood of P=0.5.  Like this:

For even more clarity, each magic paving stone is inserted between existing paving stones with equal and independent likelihood.  So, if the number of magic paving stones per round that you preset is two, and there are three paving stones, then each magic paving stone could appear in either slot #1#2 or slot #2#3, with an independent likelihood of P=0.5.  This would lead to a likelihood distribution of P=0.25 for both to appear in slot #1#2 or both in slot #2#3, and P=0.5 for one each in slots #1#2 and #2#3 (because there are two variants of this outcome as shown below).

The apparently simple question is: 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?

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Note that your only decision involves the preset number of magic paving stones that appear each round.  You don't place those stones, they just appear.  Once you set the number, that's it, you can't change it.  Since we are after the minimum number, and we are talking about magic paving stones, we could also add that if you choose a number to be the minimum that is wrong, the stones won't activate at all and the monster gets you.

Observable Events Curve - Not Quite a Drunkard's Walk

I retracted this for a while, before reinstating it.  There might be more some edits required.

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In Observable Events Curve - Is Double Dipping Essential? I showed the output of a relatively simple Excel sheet which models how a photon moves towards an observer in a universe with granular expansion.

I then got to thinking about how I could use the underlying principles to trace back the possible path(s) of an observed photon, at a given time, with different characteristics of the universe in the past.

So I wrote a script in VBA to generate a table containing sufficient data to create the chart that represents one possible path of the photon, which I could refresh as often as I liked to see what happens.

In brief, I wanted two randomly generated locations between the observer and the causal horizon (where the causal horizon is the limit beyond which a photon travelling in the direction of the observer would never reach that observer where the causal horizon is the limit [at the time] at which an object [at that distance] would recede from the observer at the speed of light).  I assumed that the reverse path of the photon would be one “grain” of additional space each “grain” of time, minus any “grains” of space that are inserted between the photon and the observer.  This might sound odd, but note that, if ran forward, the same logic would lead to a path of the photon that reduces by one “grain” of space per “grain” of time, and any expansion “grains” between the photon and the observer would increase the separation between them.

The effect is a little like a random walk (or drunkard’s walk) with the photon going through the repeated process of taking one step away and then a random number of steps (zero, one or two) back towards the observer’s location.  The likelihood of each number is determined by the separation between the observer and the photon as compared to the separation between the observer and the causal horizon at the time.  As the ratio of separations approaches zero, zero becomes increasingly more likely and as the ratio approaches one, two becomes more likely.

The script produces a table which I then charted to create a cloud of random locations at which “grains” of space are located at a given time (plotted just as points, not the lines between the points, with a brown point and a blue point for each given time), the path of the photon through spacetime (red), the OE Curve for a FUGE universe (green) and the causal horizon (grey).  For a FUGE universe that is 14 billion years old, it looks like this:

If I modify the inputs to make it a Standard Model universe – meaning that the expansion rate is initially H(t)=2/3t and then such that there is accelerated expansion post the beginning of a Dark-Energy-Dominated Era 4 billion years ago – I get this:

 

Note the problem: a photon that is observed today cannot have originated at recombination, 380,000 years after the Big Bang.  This implies that, in the Standard Model, the most ancient photon that we could possibly observe would have originated from an event about 12.27 billion years ago.

Note also that for the past 2.25 billion years, the path of the photon would have been very similar to that of a photon in a FUGE universe (within a couple of percent).

Of course, I may have made an outrageous assumption here that has resulted in me being completely incorrect, but it doesn’t seem so.  This is all very close to first principles.  Note also that I can refresh repeatedly, getting different clouds of random points and the curve does not budge.  Also, the number of points is just a display thing.  Every time, the script processes 14000 time divisions, of which I have plotted 1000 because it gets a bit busy and takes much longer when I use more.

You might wonder, what happens if I make that number larger, say 1400000?

It takes about 100 times longer to process, sure, and you get these:

 

and

 

It might be difficult to see in these images, unless you look very closely, but the effect of more time divisions is to smooth out the curves and (in the case of the FUGE universe curve) make them more closely align with the equation.  With the Standard Model universe curve, the point at which there is an intersection with the causal horizon is still approximately 12.27 billion years ago and the overall difference between the FUGE and Standard Model curves is unchanged throughout.

If anyone wants to get a copy of the macro-enabled Excel spreadsheet in order to play around with it and confirm (or refute) what I have said above, just let me know in the comments.