Counter-intuitive facts

[Kyng]

Intuition is a very powerful tool: we use it on a daily basis to 'solve' problems that we don't have the time or the patience to analyse properly. Unfortunately, while it's often a good time-saver, intuition isn't 100% fool-proof: there are times when it fails us. This is a place to list facts which are clearly true when we examine them - but which we wouldn't expect to be true if we just used our intuition. They don't need to be paradoxical - examining them shouldn't lead to any ambiguities or contradictions - but, they shouldn't be what we would first guess.

These can come from any field, but I'll start with a mathematical one:

1) Suppose you have a circle with three points on it, and you draw all the lines to connect the points. This will divide the circle into 4 regions:



Now, do the same with four points. This divides the circle into 8 regions:



Now, do the same with five points. This divides the circle into 16 regions:



You can see that the number of regions has doubled each time: we've gone from 4 to 8 to 16. We can even extrapolate this pattern backwards from three: if we had two points with one line between them, we'd have 2 regions; and if we had one point with no lines, we'd have 1 region .

So, if we do the same with six points... we'll get 32 regions?

Sadly, we don't. We get 31:



This isn't because I've drawn a wonky hexagon with unequal side lengths. In fact, if I had drawn a regular hexagon with six equal sides, then I wouldn't have had the yellow triangle in the middle - so I would have gone down to 30 regions!!!

These numbers (1, 2, 4, 8, 16, 31) are actually following a formula... but, sadly, it's a lot more complicated than "double it every time" . In case you're interested, here it is:

Formula for the number of regions

If you have N dots, then the number of regions will be:

N/24 * (N³ - 6N² + 23N - 18) + 1

So, do you have any counter-intuitive facts that you'd like to share?

[Kyng]

Here's one that really screws with my geographic intuition:

Salt Lake City is on roughly the same latitude as New York City. In fact, Salt Lake City is a smidge further north!

This just sounds completely wrong to me: when I look at the US map, I think of Utah as being "in the south-west", and New York state as being "right up in the north-east", so I'd expect NYC to be way further north. However, because Salt Lake City is right near the bit of Utah that juts upwards, and New York City is in the bit of New York state that juts downwards, and the USA curves around in such a way that the western half is a bit further north than the eastern half... they both work out being at roughly the same latitude.

[Kyng]

Another one: the closest planet to Earth is usually Mercury.

Since we're all taught "Mercury, Venus, Earth, Mars, etc...", this really screwed with my intuition: I would've thought our 'nearest planet' would surely be either Venus or Mars ! However, when those planets are on the opposite side of the Sun from us, Mercury is closer - as the following CGP Grey video demonstrates quite neatly:



In fact, it's not just Earth either. For the same reasons, Mercury is Mars's closet neighbour on average... and Jupiter's... and Saturn's.... and even Pluto's .

[Kyng]

I'm sure some people will be familiar with instant-runoff voting (IRV). Australia uses it for its elections, and the UK had a referendum on it in 2011 (where we voted against introducing it). For those of you who don't know: you rank-order the candidates (instead of just choosing one); the least popular one gets eliminated (and their votes given to the voters' second-favourite candidate); and so on until somebody has at least 50% of the vote. The idea is that it stops ideologically-similar candidates from splitting the vote, and handing a victory to their ideologically-opposed rival (as happened in several constituencies in the UK this year; IRV would have prevented these screw-ups ).

Unfortunately, IRV is vulnerable to a very unusual (and very counter-intuitive) form of tactical voting of its own: you're sometimes better off staying at home and not voting than you are voting for your preferred choices!!!

I'll demonstrate this with a fairly simple example:

1000 voters, instant-runoff voting

350 voters prefer Labour > Liberal Democrats > Conservative
200 voters prefer Liberal Democrats > Labour > Conservative
450 voters prefer Conservative > Liberal Democrats > Labour

First of all, let's suppose that everybody votes sincerely, according to their actual preferences. In this election, nobody gets 50% of the vote right away, so the least popular candidate (the Liberal Democrat) gets eliminated, and their 200 votes get transferred to Labour. Now, Labour has 550 votes (with the Conservatives on 450), so the Labour candidate wins the election.

However, let's now suppose that, instead of voting, 260 of the Conservative voters simply stayed at home. This time, the electorate would look like this:

740 voters, instant-runoff voting

350 voters prefer Labour > Liberal Democrats > Conservative
200 voters prefer Liberal Democrats > Labour > Conservative
190 voters prefer Conservative > Liberal Democrats > Labour

Once again, nobody has over 50% of the vote, so the least popular candidate (the Conservative) gets eliminated, and their votes get given to the Liberal Democrat. Now, the Liberal Democrat has 390 votes (with the Labour candidate on 350), so the Liberal Democrat wins. From the perspective of the 260 Conservative supporters who stayed at home, this is a better outcome than Labour winning - which is what they would have got if they had voted !!!

First-past-the-post voting has many problems; however, this isn't one of them. At least under that system, voting is always better than not voting .

[Kyng]

Another counter-intuitive fact about US geography, similar to the one about Salt Lake City and NYC from my second post:

Detroit is further east than Atlanta. This seems completely wrong: after all, Atlanta isn't far from the East Coast, while Detroit is a long way inland. However, because of the way in which the East Coast curves around (and gets further west the further south you go), this still puts Atlanta a smidge to the west of Detroit...

[Lurkerish Allsorts]

Another counter-intuitive fact about US geography, similar to the one about Salt Lake City and NYC from my second post:

Detroit is further east than Atlanta. This seems completely wrong: after all, Atlanta isn't far from the East Coast, while Detroit is a long way inland. However, because of the way in which the East Coast curves around (and gets further west the further south you go), this still puts Atlanta a smidge to the west of Detroit...

This didn't surprise me despite never thinking about it / the curve thing.
Turns out you can see it on even flat maps, even if its more of a subconscious thing mostly (i.e.not something you would notice normally)
Here is a straight line from the center of the dot of Detroit to the Atlanta
You can see its slightly right to the center of the Atlanta star marker.

[Kyng]

Even though most saxophones are made of brass, they aren't classified as brass instruments. Instead, they're classed as woodwind instruments.

[Kyng]

Another geographical one, which I learned today:

St John's, in Newfoundland, is closer to Croatia than it is to Vancouver. You can see for yourself on the following pair of maps:


(Source: Maps on the Web)

[Kyng]

Another one, quite similar in nature to the previous one.

The northernmost part of Brazil is closer to Canada than it is to the southernmost part of Brazil:



I know I've seen enough of these that they shouldn't surprise me any more, but they still do !

[Kyng]

Even though most saxophones are made of brass, they aren't classified as brass instruments. Instead, they're classed as woodwind instruments.

Further to this, the didgeridoo is basically the opposite: it's considered a brass instrument, even though it's made of wood!

The distinction between woodwind and brass instruments has nothing to do with the material that the instrument is made out of. Instead, it's about the manner in which the sound is produced. In a woodwind instrument, the sound is produced by dividing the air stream in two over a sharp surface (such as a reed); whereas in a brass instrument, the sound is produced directly from the musician's lips.