Two Approaches
We often hear media stories of how abysmally most of the “general public” does in basic scientific literacy tests. People don’t know that a year is how long it takes the earth to go around the sun. People think an electron is bigger than the whole atom. People don’t accept the fact of common descent, let alone evolutionary theory. There are bitter debates about what should be done and it really is a terrible situation.
Part of the problem may be how the “scientific literacy test” is presented. I think it’s the wrong format: if you’re given a list of things you SHOULD know, it creates a feeling of nagging and obligation. A bit like a list of varieties of brussel sprouts we SHOULD like. What people are told is that these things are Very Important and you’re being Very Bad by not eating your scientific greens. If you don’t already have a fascination for science, this won’t do much.
There is an advantage in that people like puzzles. Even the types of people who would not consider themselves explicitly interested in science. There’s a great online show called Scam School where Brian Brushwood (a magician) goes around US bars and shows tricks, “selling” the secret in exchange for a beer. He gets great participation from the same people that might be sneered at as the “general public” in a news report. When he gives you 6 matches and tells you to make 4 triangles, you will take it as a challenge to understand what the hell’s going on. Brushwood is NOT using the eat-your-greens method.
1, 2, 3, Infinity
I recently read One, Two, Three — Infinity. It’s considered a classic of popular science writing. Quite interesting given that it was written in 1947 (revised 1961) and so contains a lot that seems quaint. It was a great read in that he explains a lot of basic science concepts in an easy and concise way for the layman*. If anything, it was a bit too concise in places. For instance, he devotes about three paragraphs to how quantum mechanics means we should stop thinking about particles as having definite positions and trajectories. A lot of the time it feels like the concepts are mentioned so briefly you’ll only understand it if you’ve seen them before.
However the book’s great strength is that it’s one place that contains a very well edited collection of the most fascinating things in science. The topics aren’t directly presented as puzzles but from this format they can be quite naturally transformed into puzzles that challenge the reader to understand them. For example, when talking about relativity it mentions that something you experience as space, another observer might experience as time and vice versa. Now, like many newspaper headlines, it can be a bit misleading and there’s a debate about how “sensationalist” one should be. However it is a good hook — introduce something that seems so crazy that some people** will just have to find out more.
The List
Here’s a list I made, based mainly on ideas from One, Two, Three — Infinity (and a bit from Asimov’s New Guide To Science). The goal being to identify some kind of “shortlist” of fascinating concepts and to try put some of them into puzzle form. Many of these are seeming paradoxes (or even interesting facts) more than puzzles, but this is in line with the broader definition of a puzzle. And they should still all require a person to dig deeper in order to solve the puzzle or understand the fact. For some, I couldn’t think of anything so I left it blank. Let me know if I’m missing something or if you have any suggestions. Otherwise, imperfect as it is, at least it employs the sensationalism of journalism (especially science journalism) for good rather than for evil.
- [Exponential growth]: A simple puzzle with 3 sticks and 64 discs takes far longer to solve than the age of the universe.
- [Cantor]: Some infinities are bigger than others.
- [Topology]: By using some very fine cuts, you can turn 1 sphere into 2 spheres of the same size.
- [Special Relativity]: Something you experience as space, another observer might experience as time and vice versa
- [Special Relativity]: Your umbrella has no “true” length. Your life has no “true” duration.
- [General Relativity]: If you get close to the sun, the angles of a triangle do not add up to 180 degrees.
- [General Relativity]: The apple falls on Newton because it’s following the shortest 4-dimensional path possible: one that goes through Newton’s head.
- [Periodic Table]: ???
- [Statistical Mechanics]: If you sit on the couch for long enough, you’ll die from suffocation because all the air in the room will bunch up in a ceiling corner.
- [Quantum Physics]: Fundamental particles such as electrons are waves — but only when we’re not looking. When we look, they turn into particles. [This is a standard line but I think it's not too bad]
- [Quantum Physics]: Particles do not have definite positions or velocities and do not move in defined paths.
- [Many Worlds]: If you play Russian roulette with a Geiger counter such that it has a 99% chance of killing you each time you press the button, and you press it 100 times you will experience surviving all 100 blasts and going on to live the rest of your life.
- [Nuclear Physics]: If you could move subatomic particles at will, you can rearrange a piece of poo to become pure gold. [Again this might seem obvious but I for one haven't lost the wonder.]
- [Nuclear Physics]: One gram of water contains enough energy to blow up
. - [Cells]: Each of your cells is actually a symbiosis of two completely different organisms that live and reproduce side by side.
- [Basic Genetics]: ???
- [DNA]: ???
- [Cosmology]:Although nothing can move through space faster than the speed of light, the galaxies that are far enough away from us are expanding away faster than the speed of light.
- [Stars]: The light that you see coming from the sun took about 6 minutes to reach you from the sun’s surface, but anything up to a few million years to reach the sun’s surface itself.
- [Evolutionary tree]: Swimbladders in fish evolved from lungs, not vice versa.
- [Probability]: Between 24 random people, the chances of there being a double birthday are greater than 50%.
- [Incompleteness Theorem]: ???
- [Halting Problem etc]: For every program there is a definite answer to whether it will stop or go on forever for a given input. But it’s impossible to create any general procedure that will determine which of these will happen.
- [Neuroscience]: You have two personalities in one brain.
- [Psychology]: Your brain is fooling itself.
That last one is especially important!
*As an aside, I couldn’t think of a non-quaint substitute for layman; is there anything that doesn’t conjure up the image of a white male Catholic non-priest?
**Of course there’s nothing that works for everyone.
30 comments ↓
Watch me ignore the depth and breadth of your post and point out that you messed up the brackets in this sentence:
Otherwise, imperfect as it is, at least it employs the sensationalism of journalism (especially science journalism for good rather than for evil).
More on topic, your list could do with a lot more links for the curious to folow…
Follow.
Argh! Fixed.
I deliberately kept the links to a minimum so as to make the curious go forth and gather… I can’t really talk about the curiosity of puzzles whilst making all solutions 1 click away, now can I?
Why not? The curious will go on to read the links. That’s a bloody good start!
There are a few links already — are there any particular unlinked ones that peaked your interest?
“Fundamental particles such as electrons are waves — but only when we’re not looking. When we look, they turn into particles. [This is a standard line but I think it's not too bad] ”
No, depends on HOW you look. They can be viewed as waves or particles depending on the experiment. I even seem to remember a TEM picture that showed the wave state…
As I said, each of these is very simplistic and sensationalist to the point where lots of caveats are needed. For instance the basic 2 slit experiment just shows wave interference on the back screen. But I couldn’t think of a more precise way of describing it in one sentence that wasn’t too wordy.
[...] Eating Your Scientific Greens — a Nadder! anadder.com/eating-your-scientific-greens – view page – cached We often hear media stories of how abysmally most of the “general public” does in basic scientific literacy tests. People don’t know that a year is how long it takes the earth to go around the… Read moreWe often hear media stories of how abysmally most of the “general public” does in basic scientific literacy tests. People don’t know that a year is how long it takes the earth to go around the sun. People think an electron is bigger than the whole atom. People don’t accept the fact of common descent, let alone evolutionary theory. There are bitter debates about what should be done and it really is a terrible situation. Read less [...]
“[Nuclear Physics]: If you could move subatomic particles at will, you can rearrange a piece of poo to become pure gold.”
->
[Chemistry]: If you could move atomic particles at will (which we can!), you can rearrange a piece of poo to become a nice diamond, about a third the size of the poo.
1g water = 21.481 kilotons of TNT (8.9876e+13 joules)
Interesting link for your umbrella!
Another example of exponential growth would be the story of teh invention of chess where the inventor asks for one grain on the first square, two in the secon, four on the third and so on… and bankrupts the emperor.
Not bad — that makes 1g of water about the same as the Nagasaki nuclear bomb!
The umbrella link wasn’t informational but just referring to the bottom paragraph.
The chessboard story is in 1, 2, 3, Infinity as well, I just happen to prefer the Tower of Hanoi
On the gold, in terms of moving particles at will, we can break down poo into its constituents but we can’t really glue the protons and neutrons back together to form gold — doesn’t that require a supernova? I meant it’s not something readily done in a lab until such time as we have *directed* supernova-like energies available to use.
Re: Gold
yeah, but that was your example – I went with the chemistry one where we can re-arrange the carbon atoms to form diamonds – it just isn’t cost effective.
Oops, my eyes skipped right over the word diamond
The mathematical ones are the ones I find most fun, possibly because I can get my head around them the furthest and appreciate how weird some of them really are. And though people are often interested in hearing about them, I’ve never really thought of something like Cantor’s diagonal proof as something anyone should know. It’s just neat. And there could be an interesting point there, that people like neat stuff more than stuff they’re stupid for not already understanding.
By the way, my friend Marija mentioned you recently as somebody she also knows. Small world.
Cubik’s Rube: “I’ve never really thought of something like Cantor’s diagonal proof as something anyone should know.”
I beg to differ and say I WISH everybody knew something about it because it is so fundamental in understanding what kinds of infinity exist and this, itself, is fundamental for understanding what (even finite) numbers are.
Alas, the general public is too much spoon-fed with idiocies to worry about details like this.
More comments later.
Easy one on infinites:
How many number are there?
–An infinite number.
How many even numbers are there?
–An infinite number.
Are there more numbers than even numbers?
–Wow, yeah, there must be twice as many.
keddaw:
Yes, exactly. There are many ways of interpreting infinities and we have to choose the correct one, the one pertinent to our problem.
You are saying that that if E(n) is the number of even numbers smaller than n then E(n)/n converges to 1/2.
But if by enumeration we mean what the first nomads did, i.e. assign a stone to each sheep in the flock, then we can assign precisely one number (stone) to each even number and use all numbers and not leave any even number out. Therefore the number of even numbers is the same as the number of all numbers (integers).
How many real numbers are there between 0 and 1? Well, infinitely more than the number of integers.
And why do we need to understand this?
Because if we don’t then we don’t really understand Statistics, say. For there is a probability distribution, called the uniform distribution, which assigns probability L of selecting a number from within an interval of length L. But then the chance of selecting
a specific number, say the number 1 over the area of a circle of radius 1, is zero. This benign “paradox” is tantamount to understanding that there are infinitely many real numbers between 0 and 1.
Ooops…. I meant to say “infinitely more real numbers than integers between 0 and 1″.
Takis: It may be symbolic of the kind of ideas that exist which are fascinating and beautiful and under-appreciated by the general public, but I really don’t think that Cantor’s diagonal theorem specifically is for everyone. I would agree that everyone could do with some kind of appreciation of how counter-intuitive numbers can be. The birthday paradox is a fine example of that, and easily understood. And there are many other ways that statistics can be weird and do things you don’t expect – Simpson’s paradox, for instance. Being aware of some things like that might make people less sure of their own subjective judgments on things where the intuitive answer is likely to be wrong.
keddaw: Actually, the set of integers and the set of even integers are the same size. Take 1, 2, 3… and double them, and you have 2, 4, 6… There’s a direct one-to-one correspondence between each element, so the infinities there aren’t actually different sizes. It gets weirder than that before one infinite is bigger than another.
Cubik’s Rube: Yep, she’s mentioned you. But then again we come from a specific section of the blogosphere so it’s not that much of a coincidence. If you were a fundamentalist Sikh I’d be more surprised!
I think most of the items on my list are things people “should” know in the same way as an educated person “should” know about Dante, Shakespeare, Confucius etc., but it was just a matter of how this shouldness is presented. I say this from my optimism about people in that most who have a *decent* education understand concepts that are just as difficult than these. And I guess by knowing about (say) Cantor, I don’t mean being able to prove it or even follow the proof 100% but at least know about the result and what makes it interesting.
And of course maybe other things like the bday paradox are more important for practical life but I think it’s still a good aim to have most people know as many of the above as possible.
Takis: I think you can understand statistics (esp. enough so as not to get fooled by hoaxes and media sloppiness) without understanding Cantor directly.
Thanks for the comments. Maybe I’m a fundamentalist
Here’s another probability paradox, much more interesting and much more counterintuitive than the birthday paradox or the prisoner’s dilemma.
There are 1000 prisoners and 1000 boxes, each containing the name (the identity) of a prisoner. The contents are unknown to the prisoners. Each prisoner is asked to step forward and open any 500 boxes he likes. If he finds his name in one of the boxes he goes to another room and waits. The boxes are shut again (the contents are not changed though) and the next prisoner (who has absolutely no information about what the previous ones did) is asked to do the same thing: open 500 boxes, look for his name and move to the second room provided he finds it. Here is the caveat: If all the prisoners are successful in finding their names then all of them are released. But if one of the prisoners fails to find his name then all of them are immediately executed.
Question: Is there a strategy that would ensure a significantly large probability that the prisoners will all survive?
This is a mind-boggler even for professional mathematicians. The answer is easy to describe, but I will not reveal it now. Suffice to say that the worse the prisoners can do is, simply, proceed completely at random: the first prisoner has chance 500/1000 = 1/2 of finding his name; so does the second prisoner and the third and so on. Then the chance that all of them will find their names equals (1/2)^1000 = 10^(-301), approximately, by independence.
So the baffling this is: how can the prisoners beat the seemingly unavoidable independence hypothesis?
N.B. I do not claim ownership of the problem. I will reveal the source soon.
The periodic table could generally be explained with a sort of jig-saw puzzle: same groups have same colours and the elements must be arranged with increasing atomic number. This could even work for kids, as well as the plebs (how’s that for substituting layman?)
I also like the idea that if you bang your head against the wall you could theoretically pass right through it.
Takis: I couldn’t even find the name for this problem. And of course intuitively it seems like you can’t do better than (1/2)^n so looking forward to the revelation!
Shly: By groups do you mean elements with similar properties that form vertical bands on the table? Yeah, the banging the head one is similar to the one about suffocating because the air ends up in a small part of the room — and you’d probably have to bang your head a VERY large number of times to get it to pass through..
Michael: Indeed, it’s a relatively new problem and it has escaped probabilists for long time. I was told this problem by a friend of mine who does Algebra and, of course, I immediately had the same intuition as you. After thought, I realized I can do better, I thought of strategies but I could not compute their probabilities–they looked too hard. Well, the solution is *really* surprising. I love the problem because it is so mind-bogglingly counter-intuitive, much more counter-intuitive than anything I’ve seen before. I’ll write a post on it as soon as I find a little bit of time…
Oh, so there’s not even an “official” name for the problem yet? In that case I eagerly await for the post.
Hi Takis,
Just read that prisoner problem.
On first reading I think the worst they could do is NOT random, but agree to go for the same 500 boxes each. This would have a probability of execution of 1.
The best strategy (off the top of my head) would be for the prisoners to each have an individual set of boxes to open and for each of these sets to equally distribute the boxes between them.
Another strategy would be for every even prisoner to go for the even boxes and every odd prisoner to go for the odd boxes.
The probability of that working is vanishingly small, but it is better than random.
I look forward to a mathematically impressive solution.
Reminds me of that gameshow where you choose from one of 3 prize doors, the host removes one of the other doors that doesn’t have the start prize then asks you if you want to swap doors. (Most people say it’s 50/50 so it doesn’t matter – and are likely to stick with their initial choice – it isn’t and they should swap.)
keddaw: Now there is pressure on me to write something. I will. I just need a bit of time which is a scarce resource at the moment [sic]. Well, your try is good, I had similar thoughts, but the actual solution is much more impressive.
By the way, this problem is orders of magnitude more impressive, more counterintuitive than the 3-door problem (Monty hall), not least because veritable mathematicians have failed to find a solution. (Whereas, the Monty Hall problem is an exercise: once you set your mind to it you can solve it.)
I promise to write!
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Takis has posted on the 100 boxes problem and the solution’s amazing.
http://randomprocessed.blogspot.com/2010/01/totally-nontrivial-prisoners-problem.html
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