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Non-intuitive

Entry 1379, on 2012-04-20 at 21:46:01 (Rating 2, Comments)

If I gave you a piece of paper and asked you to fold it in half, then half again (so that it was then four layers thick), then again, and again, how many times would you need to fold it before it was so thick that it would reach most of the way to the Sun (100 million kilometers)?

Most people are surprised to hear that the answer is 50. In fact I have seen at least one person try it. She was convinced she could fold the paper 50 times and prove me wrong but only got to about 5 folds (because it got too awkward and the paper wasn't big enough, but you should ignore these practical difficulties for this thought experiment.)

Let's think about it. Say the paper is 1 mm thick (I know that's quite thick for paper but it makes the maths simple and doesn't really change the end result). After the first fold the paper is 2 mm thick, then 4, then 8, 16, 32, etc.

So after 5 folds the paper is over 3 cm thick. What about after another 5? After a total of 10 folds it is over 1 meter thick. After 15 it is 32 meters thick, after 20 over 1 kilometer, after 30 over 1000 kilometers, after 40 over a million, and by the time the 50th fold is completed over a billion (that's more than I said earlier because my paper started off too thick but you can see that it works in principle). That is the power of a geometric progression.

How many more folds would you need to make to double the thickness (from 1 billion to 2 billion kilometers). Some people say another 50 folds, but anyone who is paying attention will know the answer is 1. Just one more fold makes the paper a billion kilometers thicker because each fold doubles the previous thickness.

Again I must emphasise this is a thought experiment and obviously cannot be done in practice. Or can it? What about something similar involving self-replicating nano-machines?

People find this effect counter-intuitive because they tend to think linearly rather than geometrically. The expect things to increase smoothly rather than to accelerate suddenly.

Here's another example of something counter-intuitive: take a pack of cards and shuffle it (randomly) then look at the result. How likely is it that the particular order of those cards has ever existed before anywhere and at any time in the entire history of the world? There are only 52 cards so surely it is very likely that the result has occurred before, isn't it?

Well no, it's almost 100% certain that the card order you have just created is totally unique and has never existed before. The cards in a standard deck of cards can be arranged approximately 80 million trillion trillion trillion trillion trillion different ways!

The first card could be any one of 52 so there are 52 possibilities. The second can be chosen from any one of the 51 remaining, etc. So the total possibilities are 52 x 51 x 50 etc. Do the calculation and see that it gets big quickly!

The deck we use today was invented in 1480, about 530 years ago. If every person in the world (let's average that out at 5 billion) shuffled a new deck every second for all that time (about 17 billion seconds) they would still only deal 85 million trillion combinations. That is only 1 billion trillion trillion trillionths of one per cent of the total possibilities!

And you can now take a card from any place in the deck and put it somewhere else. That is (almost certainly) another totally unique combination.

People find this counter-intuitive because they don't understand statistical and probabilistic concepts.

Here's another piece of weirdness which seems trivial but maybe isn't. The phrase "eleven plus two" is an anagram of "twelve plus one". Coincidence? Most people would think so in this case because there is no political, religious, or social reason to think otherwise. But similar coincidences can often be misinterpreted as being far more meaningful when the person involved has a motive to see things that way.

People aren't good at accepting that sometimes there is no deeper reason for something and no pattern even when it looks like there is.

So what I'm trying to say here is that most people aren't good at analysing facts and coming to a valid conclusion. They aren't good at estimating the answer to numeric problems. They aren't good at establishing the probability of a particular outcome. And they aren't good at figuring out what is a real effect and what is just "noise".

It's no wonder such a large proportion of the population believe in so many wacky things like UFO encounters, psychics, gods, ghosts, lake monsters, and a bunch of other stuff. And it's no wonder they have trouble accepting scientific facts which are based on unintuitive processes such as evolution, the big bang, climate change, quantum theory, and relativity.

But those are just the obvious outcomes of this human failing. People rely on their intuition which is the result of a million years of evolution in a simple natural environment. It works well in most situations but utterly fails when trying to understand deeper meaning. The more concerning issue is that our political and economic systems assume people make good decisions. They don't.

The facts above came from another thread on Quora called "What are some of the most mind-blowing facts?" (for a similar thread see my pervious blog post.) The link for the Quora discussion thread with these and many more weird facts is here.

Comment 1 (3012) by Anonymous on 2012-04-21 at 16:00:25:

You make a big leap of logic going from some mathematical puzzles to a sort of political comment. What is that all about?

Comment 2 (3013) by RobDog on 2012-04-22 at 08:32:30:

A data transfer rate of 1587GB in about 3 seconds? Now that's something you should write about. In retrospect, I'd be happy with a tenth of a milliejac.

Comment 3 (3014) by OJB on 2012-04-22 at 14:11:43:

There are a few issues though: the data can only be transferred short distances, the operation requires a significant time interval between transfers, the receiver can often cause difficulties, once one receiver is selected it is difficult to perform data transfers to others, the data transfer protocol between sender and receiver can be highly problematic. You know, this is just too easy!

Comment 4 (3015) by Rdog on 2012-04-22 at 22:11:51:

Easy yes, but you can't tell me it wasn't fun.

I had more written, commenting in a more serious way about the above, but then after rereading it I thought "this doesn't make sense to me and I wrote it". If only more people exercised the delete key.

Comment 5 (3016) by OJB on 2012-04-23 at 14:37:34:

Here's my rule: if you are writing something which you are unsure of (maybe you think it might be too controversial, based on unsubstantiated opinion, etc) just leave it for an hour then go back and re-read it. It's amazing how often you might then want to use the delete key, that key which is underutilised by so many.
Remember: "Even a fool, when he holdeth his peace, is counted wise: and he that shutteth his lips is esteemed a man of understanding." (Proverbs 17:28), or a more contemporary version: "It is better to remain silent and be thought a fool than to open one’s mouth and remove all doubt."

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