What is Two and Two?



Here are some nice easy questions:


What is two and two?

What Is the value of g?

What colour is ferrous chloride, Fe Cl2?

Is Hong Kong in China?














Of course, you have answered four, ten, green and yes, If you have another answer, you are either (i) in error or (ii) using a different sort of precision or (iii) thinking well and are not in need of this chapter.



If I ask the question a different way – just by changing the introduction - let us see if you come up with different answers:

Suppose the questions become:

What are the possible answers to these questions:

What is two and two?

What Is the value of g?

What colour is ferrous chloride, Fe Cl?

Is Hong Kong in China?



Suppose the questions become:

Give the most correct answer you can to each of these apparently easy questions, using as much space as you need.

What is two and two?



What Is the value of g?



What colour is ferrous chloride, Fe Cl2?



Is Hong Kong in China?



Now suppose I tell you that there is an argument that says that each of four, ten, green and yes is not true.

Do you feel cheated?

Do you feel I have somehow changed the rules?

Did the question change or was it did your perception of the question that changed? What is different?

Hopefully, you see that any change you recognise is the way I want you to view these questions.



What is two and two?


The first question has been asked by me for many years and has served as a test to see if (whether) you have been unfortunate enough to have been taught by me. I even found my answer already on Wikipedia, on looking to offer a contribution.

The problem is that we assume that “two’ is an integer. That is, the assumption made is that the thing being counted is something like people, books or fingers. If you think instead of measurements, or, as a mathematician would put it, real numbers, then that answer is not four. Read on...

How tall are you to the nearest metre? Probably two metres: few of us remain under 150 centimetres and very few people are over 250 centimetres. So if you take your body length as being ‘two metres’ and you add two of these (different people; go find a friend) together, ‘two and two’ might well be nearer to three. It is, incidentally, wrong to say at this point that both of you must be under 1.75m.

How high is the frame of the nearest door to you right now? How high is the ceiling? Most doors are built to accommodate very average humans and so are built to be 220 to 230 cm high. This means the few very tall people have to be careful in doorways. There is not a lot of point in heating or cooling large volumes of space above our heads and so generally we build houses and schools and offices with ceiling heights of around 230-250 centimetres. If you are in such a building or can persuade yourself to think of  being in one, then each floor to ceiling height is, yet again, two metres to the nearest metre and two of these added together come to 460-500 centimetres. Which is nearer five metres than four. So if we had taken a length or height of 225 cm to just short of 250 (because the convention when approximating to one significant figure is to round 250 up to 300) then these ‘twos’ come to five.


The appropriate word of criticism, for those who dislike this sort of thinking, is ‘pedant’. Another suitable word is mathematician. The point is that you begin to question what was meant by the counting word ‘two’.


Writing the answer in mathematical language:

If two is taken as an real approximation of an unknown quantity, x, to one significant figure, then  1.5 ≤ x < 2.5,  so  3.0 ≤ 2x < 5.0. Putting this back into words, two and two could be anywhere between three and five, depending how you view the ‘two’ in the question.

And, indeed, that doesn’t include thinking further about what you were counting. If we were adding things that don’t express size, one and one can be one: if you join two crowds of people you have a bigger crowd; the same applies to digging holes or to joining rivers or, perhaps, counting debt. So, if my question had been

How many things do you have if you have two of them ?

then, apart from thinking it a pretty stupid question, you might now see that answers depend greatly on what the thing is, and generally the answer would be one or two, but not three because ‘thing’ is a word we use as a general, unspecific amount. The number ‘two’, on the other hand, expresses a precision – but unless we state that precision we are making assumptions.


What is the value of g?


This looks like a Physics question. Most people will answer ‘ten’ and slightly better students add the units, metres per second per second. They are correct to only the first significant figure. An elderly Briton or a North American might tell you the answer was 32, which is in feet per second per second.

The constant g is the acceleration due to gravity measured at the surface of the planet Earth (or Terra, but meaning this planet we live on). A non-italic g represents a gramme, though usually we would pick this up from context.

The value of g varies with the local density of rock and with the distance from the centre of the planet (properly, the distance from the centre of mass of the planet, which is only roughly in the middle). The dense material inside our planet is not stationary and not uniform in density, so the value of g as experienced on the surface is not a constant. However, it is extremely rare for the figure in m/s/s to rise as high as 9.9 or to fall to 9.7. In Britain, a figure of 9.806 is a commonly used value; if you use the argument from the previous question, then 9.8055 ≤ ‘9.806’ < 9.8065 and this value applies to most of the British Isles, exceptions being in the close proximity of high concentrations of denser rock. Such as Cheviot in Northumberland or Castle Rock in Edinburgh.

Good students of Physics should ask themselves how high you need to fly to make g no longer correct at 10 (so, how high do you have to be for a local measurement of g to be less than 9.5 m/s/s?) Similarly, one should wonder how deep you need to be before g is actually 10 to two significant figures (at least 9.95). These good students would also wonder how measurements of g change near volcanoes, because we know that the molten rock (magma) inside the volcano has come from deep in the planet so is probably made of denser material, which would affect the value of g. Vulcanologists (specialists in the study of volcanoes) use the local measurement of g to five or six figures to map the movement of the magma and they use maps of the changes to attempt to predict eruptions.


So this question has several better answers:

a) 10 metres per second per second to one significant figure b) 9.8 m/s/s to two significant figures c) 9.81 ms-2 to three significant figures. I leave aside whether you think the units are necessary when the question asked for the value of g: Physicists generally insist on adding the unit whether mathematicians happily separate the unit from the number. Many school students give that answer that their particular exam board uses. They often confuse the convenience of using g=10 with the precision they give in their answers – since 10 is only correct to one significant figure and not to two, their answers to extended calculation are only correct to the same precision. This is a source of much confusion and one must applaud exam boards such as MEI that declare g to be ‘exactly 9.8 m/s/s” which avoids all the rounding issues. Similarly, boards that state g to be “9.81 ms-2 to three significant figures” leave subsequent recognition of the use of rounding so that it falls within other examination rules that fix three significant figures as a requirement, one that is not in conflict with the value of the constant given. Far too many students treat 9.81 as being an exact value.


So the value of g requires a statement of precision. The answer ‘ten’ is actually not a correct value at anything but one significant figure; this is very unhelpful as it means that 9.5 ≤ g < 15 (really, anything less than 15 rounds to ten to one sig.fig). Continuing to comment without adding the units, the assumption that 10 means 9.5 ≤ g < 10.5   - which is reading ten as correct to the nearest whole unit - is just that, an assumption. Many students then incorrectly assume that values of g are scattered evenly across this interval, when there are smaller intervals which correctly describe the distribution of values of g.


The value of g varies with your location on the surface of the Earth. Numbers in the range 9.75 ≤ g < 9.85 are correct in all cases so far explored. The mean across the planet is taken as 9.806 to the precision implied by that number. There is a lot of sense in using 9.81. There is a ‘standard gravity’, gn = 9.80665. There’s a map of the globe showing contours of equal gravity here and a distorted rotating  geoid here.


What colour is ferrous chloride ?

The textbooks in China say this is green. If you make some in the chemistry lab it is a dirty white colour with a tinge of green. If someone is ill, we sometimes say they are ‘looking green’ – we recognise tinges of green in the otherwise usual pinkish brown that we humans generally are.

Humans are very sensitive to colour, able to distinguish a large number of colours, in the millions. The researchers in paint supply tell us that we recognise thousands of different colours and that we can distinguish something like 1100 different shades of white. We have a problem in describing colour. You cannot guarantee that what you call green is what I call green; while we might agree that new grass is green, we are unlikely to agree on the range of colours that we are prepared to call ‘green’. It is a problem of description.

If you make ferrous chloride from extremely high quality materials, what the chemists would call reagent quality, then the colour produced is much more strongly green, but it is still a green heavily mixed with white. As the materials become contaminated with traces of others, so the shade of green rapidly fades and ‘white’ is the colour seen.

In China we have a problem in arguing with text books – a matter of which I write at length elsewhere – and because the book says it is green we say it is green even when it clearly is not. <much material on the ‘Chinese issue’ here suppressed for writing about some other time> There are problems with practical chemistry (and with chemistry practicals) in recording what you actually see and recognising what you are expected to see. In examination conditions the wiser and better-taught student will record what they actually see and what they expected to see. “The resulting mixture was white with a tinge of green. This may be the green of ferrous chloride but my test is inconclusive and unconvincing” is a typically honest and accurate statement.

The form of the iron (II) chloride most commonly seen in the laboratory and in commercial uses is the tetrahydrate, formed from crystallisation from water by treating an appropriate ferrous salt with hydrochloric acid, HCl. Solid, anhydrous Fe Cl2 is a tan colour; the solid anhydrous ditetrahydrate is a pale green. Yes, I looked that up.

Iron (III) chloride, ferric chloride, Fe Cl3, is brown in aqueous solution. For students of chemistry the distinction between the ‘green’ and the brown is the significant test for establishing whether they have iron (II) or (III), the ferrous or ferric forms of a salt under test.

So the fundamental problems with this question are:

(i)  Was there an assumption that the ferrous chloride in the question is the form in aqueous solution?

(ii  )Do we agree to call this colour green?

(iii)      Are we prepared to call this colour green when white is a more accurate description?

In many ways, our underlying problem remains the same: what do we mean by the words we use?


Is Hong Kong In China?

Well, you would think so. In 1999 Hong Kong reverted from being a British-run protectorate to being part of China. It remains a semi-autonomous zone and is an administrative district or province in its own right. Technically,  it is a Special Administrative Region, an SAR; as is Macao, which means that the PRC central government allows a high degree of autonomy, excepting, understandably, diplomatic relations and defence. It runs its own currency, the Hong Kong dollar, only roughly on a par with the renminbi. Cars drive on the left, as a vestige of the British governance. The legal system is quite different from the rest of China. You must show your passport at the border and go through customs and health checks each time you cross in and out. If you fly into HK then you need a visa to go onto the Mainland. Chinese residents need a visa specifically for travel to Hong Kong; Hong Kong residents have their own passports, the HKSAR passport. National government comes from Beijing. Local and district government comes from within the city. As a Briton, I have, indeed, less hassle getting into HK than a chinese does; I don’t ned a work permit, either. The parliamentary system in HK, the HK dollar and the legal system will last until 2049.

So, in many senses, Hong Kong is less a part of China than Tibet is and, to a foreigner, quite as separate as Japan. Or Taiwan. Or the Philippines, or Britain. If you are Chinese, it is thought to be easier (quicker, less hassle, more likely to be granted) to apply for a HongKong visa than for elsewhere, but several I asked said that in some ways it is actually easier to go to the Philippines because the visa can be applied for and granted on arrival at Manila airport, which makes it actually easier to go to.

The official language of China is Mandarin: the official languages of Hong Kong are English and Cantonese.

Ask someone in the street in HongKong (the best test, in my view), and they will say they are in China. I read the situation as an expression of the will of China to run its shopfront as effectively as possible.


Is Hong Kong in China? At one level, that of the government, indisputably so,. At many levels that affect the individual, Hong Kong is as much outside China as Chinatown in San Francisco. Oh, and my bank, the HongKong and Shanghai Banking Corporation, is absolutely clear on this: Hong Kong is not in China. I have the same problem with using any bank card in Hong Kong as I would in Honolulu, Hanoi and Hannover: not much at all.

© David Scoins 2017