15 Prisoners – Solution

The prisoner puzzle is apparently annoying other people as much as it did me, which is, you know, gratifying. It was a present when I was a kid, and became one of those items that pops up intermittently over the years, annoying me more each time. I confess I never figured it out, and I eventually stumbled across an explanation online. There are a few different versions out there, all using the same principle.

So, what’s happening to the extra prisoner? He’s being absorbed into all the others.

Look carefully at the remaining prisoners and you’ll see that each gains a small amount of mass – a bit of leg here, a forehead there. The puzzle is cleverly designed so that the individual prisoners appear to remain consistent between turns, but actually each body shape is duplicated, so they’re not ‘the same’ people after the turn. But, the duplicates are slightly different sizes. Check out Bushy Eyebrow guy just next to the ‘A’, going clockwise. When the wheel is at ‘B’ his legs are just that much longer than at ‘A’, and indeed all of the prisoners are slightly bigger. The extra guy is being chopped up into lots of little pieces and distributed amongst the rest. Cannibal prisoners.

Still don’t like it. I can understand it intellectually, just about, but I can’t grasp it properly. Writing this out has helped a little, but I still intuitively dislike that I can’t ‘see’ the guy disappear. It’s a good one, though – I enjoy breaking my brain.

If my explanation hasn’t helped, there’s a particularly clear description here, and moillusions has further examples.

15 Prisoners

Here’s a puzzle that’s been on my wall since I was eight. 15 prisoners:

Annoying optical illusion - 2

Rotate the upper surface and…

Annoying optical illusion - 2

14 prisoners. Where did he go? Click on the images for larger versions.

I hate this puzzle. Even knowing the answer, I simply cannot get an intuitive grasp on it. I’ll post the explanation tomorrow.

Uri Geller’s ‘Phenomenon’ prediction

The US tv network NBC recently broadcast a show called ‘Phenomenon’1, a reality show which purported to look for the country’s best mentalist. The judges were: ardent skeptic and magician Criss Angel, and Uri Geller. It was hotly discussed on the skeptical blogs due to its deliberate vagueness on whether the contestants were magicians or had real psychic powers – Geller obviously claims to be looking for the real thing, while Angel was quite the opposite.

By all accounts the latter dominated the show from the second episode onwards, when a contestant claimed to be able to talk to the dead. Geller was convinced. Shock. So, on live TV Angel produced an envelope and offered $1million cash to either if they could tell him the contents. Of course neither could, and the contestant reacted in the usual way – he was offended, angry and confrontational. This was funny (and possibly a little staged).

In the final episode Angel revealed the contents of the envelope, and some have claimed that Geller correctly predicted the contents:

Did you find that a little more impressive than expected? Yeah, me too. But watch it again – it’s a great example of how memory lies.

It’s initially impressive – he definitely seemed to mention the numbers nine and one. What are the odds of that? Well, despite the text, Geller doesn’t just mention ‘1’ and ’19’. If you take just the numbers, in order it’s ‘1’, ’20’, ’19’, ’40’, ‘1’. Here’s what you have to do to get to ‘911’:

  • Arbitrarily choose the 19 as an important number.
  • It’s backwards whichever way, so reverse it, for no reason.
  • You need another 1, so choose just one of the two mentions of ‘1’. Maybe he mentioned it twice because it’s important, or something. Put it afterwards, even though none directly follow it.
  • Stop, for no reason.

There’s no logic there, nothing replicable; nobody looking just at those numbers would come up with an obvious meaning of ‘911’. There are a huge number of possible numeric combinations if you allow the above machinations – ‘911’ is clearly working backwards from the result, which isn’t allowed. It’s a hell of a stretch to get from these numbers to ‘911’, but classic numerology. There are only ten digits – there’s always some way to manipulate numbers to get the desired result.

Angel happened to have written a number in the envelope. But if he hadn’t there’s still plenty that could have been used to retrospectively claim Geller was right. He mentions december, months generally, days, births and spoon-bending, none of which is relevant. He doesn’t doesn’t articulate himself very well, but if you catch his final sentence he appears to be making some kind of point about his 40 years of success, and this seems to be his main aim – there’s no hint that he’s trying to make a prediction. And seriously, you’re telling me that Geller, master showman, really knew the answer and chose to reveal it in a bunch of garble?

Plenty of YouTube commenters use a telling phrase when they call it ‘interesting’. That’s a desperate word. Even the most enthusiastic don’t think there’s any positive, smoking-gun evidence here, but it’s still apparently suggestive of something, although nobody can say what. And that’s the point. When working backwards, it’s always possible to infer odd goings-on. A few years ago The Bible Code claimed that incredibly specific predictions could be found in the Bible by laying out all the letters, picking one to start with and moving up, down, left, right and diagonally, looking for phrases. This produced astonishing results, like the ‘assassination of Yitzhak Rabin was in close proximity to letters spelling out his name’. But it worked backwards from a desired result. When heavily criticised the author said ‘[w]hen my critics find a message about the assassination of a prime minister encrypted in Moby-Dick, I’ll believe them.’. So they did, producing predictions of the Kennedy, Martin Luther King and Yithak Rabin assassinations, as well as the death of Diana, Princess of Wales. Argument destroyed.

Geller did not say anything specific – everything is interpreted after the fact. He didn’t make any claim about the contents of the envelope, nor did he make any claim about his intention to predict, or how his prediction would work. This doesn’t make it ‘interesting’, this makes it ‘ambiguous’. There’s nothing to latch onto without making unjustified claims – Geller said something that, after rather a lot of manipulation, could be interpreted positively; why, when the most obvious solution would just be to say the answer? There’s no answer to this, so you have to make something up. In this case, I guess it’s that he was picking up the correct answer subconsciously. It’s a red flag when this kind of extra supposition is necessary. Based on this extremely fuzzy evidence, which is more likely: psychic powers exist, or it’s a mixture of chance and backwards thinking?

There’s also the possibility that Geller, being an extremely capable con-man, was deliberately trying to load his comments with ambiguous phrasing in the hope of getting a random hit. I personally doubt this, as I think he’d have done a better job, but it’s always possible – the guy’s spoon-bending was debunked 30 years ago, yet he’s still raking in the money from the same old shtick. Maybe he’s more canny than I suspect. If he’d scored a miss nobody would have cared, and he could have easily dismissed it as due to Angel’s attitude, but a hit of any kind (no matter how ridiculous) is going to go down well amongst his fan base.

It’s clearly the work of various logical fallacies, but even if you were to grant all these at the very best you’ve only got a hypothesis – it’s not evidence for anything. By his own admission Geller’s been doing this for 40 years, so he must know how it “works” – why doesn’t he take James Randi’s $1 million challenge and give the money to charity? Why doesn’t he get himself a Nobel prize, and change the world overnight? Strange, that.

I’ve read the skeptical literature on this kind of thing. I know about Geller’s con-artistry and his techniques. I know about numerological chances, cold reading, and the intricacy and allusion-filled-nature of language. Yet still I found it impressive on first viewing. I want to believe that psychic powers are real and my natural instinct is to latch onto something that suggests it, but I have to engage my brain if I want to avoid being tripped up. I’m a bit of a rubbish skeptic in this respect, as it takes me a while to get into the right frame of mind. Fun, though.

The show solved one mystery, albeit not a very pressing one: we now know what happened to Tim Vincent.

  1. etc. []

Faith, science, and David Hume

Dr. Steven Novella today has a good post on the ‘faith’ involved in science. He takes on the common accusation that the assumption of a rational universe is an underlying, unjustified assumption of science1:

Let us conduct a thought experiment. If we do live in a naturalistic world that predictably follows its own laws, then empirical hypothesis testing should be able to, over time, work out those laws and how the universe works. There would be no theoretical reason why science could not eventually understand any natural process. So far all the evidence seems to be pointing to the conclusion that we live in this type of universe.

What if, however, we lived in a “paranormal” universe – meaning that there were phenomena that did not follow naturalistic laws. Or perhaps our universe is somehow embedded in a grander universe that lies outside out ability to examine scientifically, but can occasionally intrude into our world. In other words, perhaps our reality, the reality to which we have access, is only a tiny slice of ultimate reality. Therefore, while we can only examine the tiny slice in which we live, it is subject to phenomena outside of that slice but part of the grander reality.

In such a paranormal universe, we would still only have science as a way to examine the world. Science could still mostly work. However, we would encounter phenomena that would not yield to scientific examination – that could not be explained or understood no matter how hard we tried. Centuries, even millennia, of examination would not penetrate these mysteries. They would forever lie outside the methodology of science as enduring anomalies.

Either way, the scientific method is the only game in town. There’s no need to assume the naturalistic universe for science to work, and in either universe science would give us some indication of which we inhabit.

The only ‘faith’ that I know of in science was pointed out by David Hume, and that’s the broken nature of inductive reasoning. As Stephen Law puts it, why should we expect the sun to rise tomorrow? Inductive reasoning says that the laws of motion have been constant and there’s no reason to think they’ll change. But that’s begging the question – Hume says that arguments from experience will always produce a circular argument, and claims to anything other than experience always use inductive reasoning. It’s a supremely irritating bit of argument.

The above still applies, in a way – science doesn’t actually claim things will always be the same, and if things changed science would simply examine what was different, but in practice science is probably wedded to the idea that results are repeatable. Anything else is practically unmanageable. Of course religion – always the opposing side in this kind of argument – suffers from exactly the same problem, no matter how much it plays around with the definition of its deity.

I’ve only been reading about this recently, and I’m sure Hume’s views are more nuanced than I’ve suggested. I also don’t know the responses of modern philosophy, but it’s certainly a fun one to think about.

  1. I like that this argument apparently accepts the pejorative sense of ‘faith’ []


I occasionally get caught out by straw-man arguments. A subtle mangling of my position and I find myself defending something I don’t believe. I’m much more aware of this tactic now, and notice it more than I used to. Sometimes, however, it doesn’t take much. Right after Ahmadinejad’s now-infamous “we don’t have homosexuals in our country”1, he said:

Maybe you think that being a woman is a crime. It’s not a crime to be a woman.

Ha. Take that, America.

I don’t know about the rights and wrongs of inviting total dickheads to speak at universities. They shouldn’t be stopped from doing so, of course, but I can see the argument that it gives them a platform. The issue tends to rest on their public speaking skills – there’s also always the danger of eloquent and charismatic speakers giving a good impression and swaying at least some of their audience. Not thinking of anyone in particular. Happily, it appears that Iran’s president is simply rubbish, and is getting the drubbing he deserves.

  1. I love how even the translator can’t help a mocking tone []

The Enemies of Reason – Part One

I really enjoyed it. I was a little worried it might just be a look at the bizarre things people believe, which would probably have been entertaining enough, but there was also an excellent explanation of the reasons we all stumble into supersition – I thought of Skinner’s pigeons seconds before they turned up1 – and probably the best tv explanation of the scientific method I’ve seen. Rather than being overwhelmingly negative about the reach of the paranormal into society, there was a healthy dose of wonder: ‘science is the poetry of the universe’.

I particularly enjoyed RD2 calling out the cold-reading card guy, and the discussion with the magical-thinking astrologer was very revealing – the guy refused to validate anything he believed! I was also amused that Jonathan Cainer’s name was blurred, but it was obvious anyway 🙂 I liked the swipe at postmodern/relativists – Melanie Phillips take note – and the dowsers, while demonstrated to be completely wrong, were treated humanely. I’d be interested to see more of the interview with the “I’ll be around for billions of years” spiritualist, although it was probably all as bonkers as the clip we saw.

I didn’t think the Warwick Uni sociologist’s point was followed up as well as it could have been. He claimed that people could interpret evidence in different ways, and that a stalemate could result. I can see that refuting this is fairly complex, though. Do you go with the concept of a scientific consensus, or argue that interpretations cannot be inherently opposed if taken from the same data – that any differences must be resolveable through logic3?

There could possibly have been more time spent on the reasons irrational belief is bad for society, but I suspect that’ll come in next week’s show on alternative medicine and the NHS. Astrology and psychics irritate and worry me, especially when you realise how much money they bilk from gullible-but-often-desperate victims, but it’s alternative medicine that’s the really despicable, dangerous area. Looking forward to it.

(update: this Richard & Judy interview sums up some of the main points. Richard M talks sense, Judy seems…less impressed)

  1. I must have read about them in a Dawkins book, I guess []
  2. who channel 4 seem to have decided is ‘Mr Grumpy Face’, given the not particularly representative pics they’re using on their website []
  3. which I’m not claiming is necessarily true, but seems possible []

Melanie Phillips on pseudoscience and how the world is broken

I’d planned the post in my head. I was going to talk about Richard Dawkins’ new Channel 4 show: The Enemies of Reason. The Telegraph describes it with:

The 66-year-old scientist has investigated a range of gurus and therapists, including faith healers, psychic mediums, angel therapists, “aura photographers”, astrologers, Tarot card readers and water diviners, and concluded that Britain is gripped by “an epidemic of superstitious thinking”.

I was going to predict responses to the show. I reckoned there’d be a couple of types. Comment Is Free might have a few “science is a faith and doesn’t have all the answers and there’s actually something to all this stuff”, and the Guardian itself would have “yes of course it’s all nonsense, but don’t you see that it makes people happy and it’s a bit mean to attack it. Also Richard Dawkins is a fundamentalist and the show would be better presented by someone else”. But I wasn’t quick enough: Melanie Phillips got in there first1.

I know she’s usually a bit, um, extreme, but this is just nuttery of the highest order. And it starts off so well:

In a TV programme to be shown later this month, Dawkins looks at a range of ludicrous therapies and gurus, including faith healers, psychic mediums, ‘angel therapists’, ‘aura photographers’, astrologers and others. Not surprisingly, he is horrified by such widespread irrationality, not to mention an exploitative industry that fleeces people while encouraging them to run away from reality.

He is right to be alarmed. What previously belonged to the province of the quack and the charlatan has become mainstream. The NHS provides funding for shamans, while the NHS Directory for Alternative and Complementary Medicine promotes ‘dowsers’, ‘flower therapists’ and ‘crystal healers’.

She agrees! Wow. I was expecting the first type of response.

Disturbing indeed. But where Dawkins goes wrong[…]

Right, here we go.

But where Dawkins goes wrong is to assume this is all as irrational as believing in God. The truth is that it is the collapse of religious faith that has prompted the rise of such irrationality.

What? Seems like a non-sequitur, but whatever. The collapse of religious faith is to blame for the rise in irrationality? This seems immediately unlikely as much of the irrationality has been around for a long, long time. The murder of Abraham Lincoln prompted massive conspiracy theories. Astrology has been around for centuries. Alternative medicine could only really be seen for what it is once evidence-based medical science came into being, but would seem to be far more in response to that than anything religious. In Britain religious faith is down, but it’s had a massive resurgence in the US, which is also a major stronghold for all types of the irrationality being discussed. So I’m not sure the timeline really works. But let’s see how she backs this up…

We are living in a scientific, largely postreligious age in which faith is presented as unscientific superstition. Yet paradoxically, we have replaced such faith by belief in demonstrable nonsense. It was GK Chesterton who famously quipped that ‘when people stop believing in God, they don’t believe in nothing — they believe in anything.’ So it has proved. But how did it happen?

Proof by repeating yourself, apparently. All right then, how did it happen?

The big mistake is to see religion and reason as polar opposites. They are not. In fact, reason is intrinsic to the Judeo-Christian tradition.

The Bible provides a picture of a rational Creator and an orderly universe — which, accordingly, provided the template for the exercise of reason and the development of science.

So, let’s get this straight. The whole world has stopped believing in god, apparently. Everybody sees religion and reason as opposites, so they’ve taken up irrational things in its stead, despite having rejected religion for rational reasons. I’m not really following this. But, anyway, it’s not even true because religion and reason aren’t opposites. We know this because it says so in a magic book, and we should believe anything written in magic books.

Dawkins pours particular scorn on the Biblical miracles which don’t correspond to scientific reality. But religious believers have different ways of regarding those events, with many seeing them as either metaphors or as natural occurrences which were invested with a greater significance.

I wonder if she’s been reading Alister McGrath – he’s always going on about ‘significance’. Still not sure what her point is. Magic book says things happened. Dawkins says they probably didn’t. Melanie Phillips says they didn’t and are of course metaphors. So? Presumably she doesn’t deny all the miracles – virgin births, a child of a god, resurrection etc. etc.? If she denies it all, she has little in common with most Christians I’ve read. She’s using the initially-persuasive idea that the Bible can be interpreted in such a way as to make logical sense. Which still doesn’t mean it’s true, but would be a start. Sam Harris and others would argue that the Bible is such a mess of contradictions that there’s no way to interpret it without simply ignoring the parts you don’t like. But I digress.

The heart of the Judeo-Christian tradition is the belief in the concept of truth, which gives rise to reason. But our postreligious age has proclaimed that there is no such thing as objective truth, only what is ‘true for me’.

Knew we’d get to relativism eventually. Note that Dawkins isn’t mentioned here. Not one of the ‘New Atheists’/’Fundamentalist Atheists’/whatever has any truck with relativism. Nor do the vast majority of scientists, as far as I’m aware. I never understand how people so willing to read Christian theology can be so ignorant of secular philosophy, which pretty much rejects relativism outright. I also strongly doubt that any sizeable percentage of the population think there’s no such thing as objective truth (outside of postmodernism students, anyway), but then I can’t really back that up.

That is because our society won’t put up with anything which gets in the way of ‘what I want’. How we feel about things has become all-important. So reason has been knocked off its perch by emotion, and thinking has been replaced by feelings.

This has meant our society can no longer distinguish between truth and lies by using evidence and logic. And this collapse of objective truth has, in turn, come to undermine science itself which is playing a role for which it is not fitted.

What? Scientists now don’t believe in objective truth, so science doesn’t work any more? What? I’m not a sociologist, but I’m pretty sure all her statements about society are complete nonsense.

When science first developed in the West, it thought of itself merely as a tool to explore the natural world. It did not pour scorn upon religion; indeed, scientists were overwhelmingly religious believers (as many still are).

Oh, for crying out loud. Yes, Newton was religious. With the information he had, it made sense. Before the theory of evolution came along it was pretty damned hard to see any other explanation. But now, with the evidence we have, religious belief is undoubtedly irrational. If Newton were around today, it’s reasonable to think he wouldn’t be religious.

In modern times, however, science has given rise to ’scientism’, the belief that science can answer all the questions of human existence. This is not so. Science cannot explain the origin of the universe. Yet it now presumes to do so and as a result it has descended into irrationality.

No it doesn’t. That’s just not true. There are plenty of questions on which science hands over to philosophy. There are incredibly speculative ideas as to how the universe started, sure, but nobody with scientific credibility claims to have actually explained it. I don’t think it’s necessarily a question outside of science, though. We just don’t know. Presumably she doesn’t mean ‘how the universe started’, she means ‘why there’s something rather than nothing’, but the same applies.

The most conspicuous example of this is provided by Dawkins himself, who breaks the rules of scientific evidence by seeking to claim that Darwin’s theory of evolution — which sought to explain how complex organisms evolved through random natural selection — also accounts for the origin of life itself.

No he doesn’t. This is also completely false. In fact he specifically says that evolution doesn’t account for that. Biochemistry is investigating that particular problem. It depends what she means by ‘the origin of life’, of course. Does she mean consciousness? Cells? Things that evolve?

There is no evidence for this whatever and no logic to it. After all, if people say God could not have created the universe because this gives rise to the question ‘Who created God?’, it follows that if scientists say the universe started with a big bang, this prompts the further question ‘What created the bang?’ Indeed, if the origin of life were truly spontaneous, this would constitute what religious people would call a miracle. Accordingly, this claim in itself resembles not so much science as the superstition that Dawkins derides.

I’m not sure she isn’t confusing the origin of the universe with the origin of life, but whatever. It might be that the origin of life is extremely unlikely – indeed, it seems that it took millions and millions of years for (presumably) one chance event to occur – but that’s not ‘spontaneous’ any more than the weather is ‘spontaneous’.

Moreover, since science essentially takes us wherever the evidence leads, the findings of more than 50 years of DNA research — which have revealed the almost unbelievable complexity of the arrangements which are needed to produce life — have thrown into doubt the theory that life emerged spontaneously in a random universe.

Uh oh. She’s not going to…she wouldn’t, would she?

These findings have given rise to a school of scientists promoting the theory of Intelligent Design, which suggests that some force embodying purpose and foresight lay behind the origin of the universe.

She did. I don’t believe it.

While this theory is, of course, open to vigorous counter-argument, people such as Prof Dawkins and others have gone to great lengths to stop it being advanced at all, on the grounds that it denies scientific evidence such as the fossil record and is therefore worthless.

A bit, but not really. The problem with intelligent design is that it’s not science. It makes no predictions. It has no causal mechanisms. It hinges completely on the idea that if evolution is wrong, god must have done it. It occupies the infinite space of crap-I-made-up-ness. I could say that the process of evolution is actually controlled by an intelligent and incredibly tiny bumblebee named Gordon. It’s possible, but a) if evolutionary theory is wrong, it doesn’t mean Gordon is real, and b) until I can provide any kind of experiment that would provide a different outcome for evolution vs. Gordon’s Design, how can we know? There are an infinite number of things that could be true, and we believe what the evidence suggests and nothing more. The reason scientists and rational thinkers have tried to stop intelligent design progressing is that it has no substance.

Yet distinguished scientists have been hounded and their careers jeopardised for arguing that the fossil record has got a giant hole in it. Some 570 million years ago, in a period known as the Cambrian Explosion, most forms of complex animal life emerged seemingly without any evolutionary trail. These scientists argue that only ‘rational agents’ could have possessed the ability to design and organise such complex systems.

Oh, man. There are any number of books which explain the Cambrian explosion. It’s actually really, really cool. I’m surprised she didn’t bring up punctuated equilibrium, but then she has just claimed all scientists are incapable of performing science. I like how she mentions the Cambrian problem, then tries to get out of it:

Whether or not they are right (and I don’t know), their scientific argument about the absence of evidence to support the claim that life spontaneously created itself is being stifled — on the totally perverse grounds that this argument does not conform to the rules of science which require evidence to support a theory.

There is no such claim, so their argument is bogus. You don’t need to be a scientist to understand this point.

As a result of such arrogance, the West — the crucible of reason — is turning the clock back to a pre-modern age of obscurantism, dogma and secular witch-hunts. Far from upholding reason, science itself has become unreasonable.

And thus, the whole of science is now ‘unreasonable’ because of, even from her viewpoint, a spat limited to evolutionary theory.

So when Prof Dawkins fulminates against ‘new age’ irrationality, it is the image of pots and kettles that comes irresistibly to mind.

Aha! I knew it!

So: the world went all rational and rejected religion. Religion, though, is secretly rational, and people are therefore rejecting rationality. So they now believe in all sorts of crap. This breaks science, because all scientists no longer believe in objective truth and think they can explain everything without using any kind of logic. This results in heroic evolution-deniers being silenced by conspiracies. Yes, looking at this evidence it does seem like religious belief lends itself to rational thinking. Also, Richard Dawkins is wrong about everything, and the program would better be presented by someone else.

I know it was fish in a barrel. I know I probably shouldn’t pay attention to such nonsense. But it was an incredibly annoying fish.

  1. I wrote all the below before showing the article to my girlfriend, who said ‘yeah, it’s Melanie Phillips’. Which is a fair point. But as it’s written I might as well publish 🙂 []

The Reasoner

The Reasoner sounds like it could be an interesting online magazine:

The Reasoner is a monthly digest highlighting exciting new research on reasoning and interesting new arguments. It is interdisciplinary, covering research in, e.g., philosophy, logic, AI, statistics, cognitive science, law, psychology, mathematics and the sciences.

The first issue contains a discussion of logical truth based around the example of Lois Lane kissing Clark Kent / Superman. I like it already.

Cannonball logic

Following on from the last post, the podcast also has a logic puzzle, and last week’s question was:

You’re in a boat that’s floating in a pond. You’re holding a cannonball. If you drop the cannonball over the side, will the level of the pond rise, fall or remain the same?

Mr Thinks-he-has-a-good-knowledge-of-physics here got it completely wrong, because I didn’t think it through properly. There’s no trick – cannonballs don’t float or anything – it’s pure science. What do you think? Highlight below for the answer.

Answer: The level of the pond goes down. When in the boat the weight of the cannonball displaces a certain amount of water. But when in the water it’s only the volume that matters – water doesn’t care about the density (and therefore the mass, and therefore the weight) of what’s in it, just the space it takes up. A cannonball is obviously more dense than water, so the volume displaces less water than the weight, so the water level goes down. I fully expected the level to stay the same, thinking that the displacement wouldn’t change. If not that, then maybe it’d go up. Going down was my last choice by a long way.