Spoiler: 2016 as the sum of 3 squares, by mental arithmetic

Dave Radcliffe  @daveinstpaul   twitted:

\(2016\) is the sum of four squares. This exceptional event occurs only \(100\) times each century.

I commented:

It is a good idea to start the New Year Day by finding these four squares. This year, it is easy. And 3 squares suffice.

and added:

This year, finding the four squares can be done by mental arithmetic Honest! Try!

So, here comes a spoiler, intentionally written with minimal mathematics notation from what I first did entirely by mental arithmetic. Indeed, observe that

  •  \(2016= 2000 + 16\) and that \(16 = 4^2\);
  • \(2000\) is divisible by \(16\) because \(2000\) is \(2 \times 10^3\), hence \(2000 = 2 \times 2^3 \times 5^3 = 2 \times 8 \times 125 = 16 \times 125\);
  • hence taking out  \(16\) out of  \(2000 +16\) simplifies the problem;
  • now \(2016 =  16 \times (125 + 1)  = 16 \times 126 = 4^2 \times 126\);
  • all that remains to do is to write \(126\) as the the sum of four or less squares and then multiply each of them by \(4^2\).

Here we start trying our luck.

  • The largest square smaller that \(126\) is \(9^2 = 81\), and \(126 = 9^2 + 45\).
  • Similarly, \( 45 = 6^2 + 9 = 6^2 + 3^2\)
  • Ha! Now  \(126 = 9^2+ 6^2 +3^2\). Multiply everything by \(4^2\) and we get \(2016 = 36^2 + 24^2 + 12^2\).

So that was what I did by mental arithmetic.

However, mental arithmetic is not optimal way of solving. In calculations above I made an error and an omission which, fortunately, were not lethal, but which I noticed only now, while writing up my mental solution.

  • \(9^2 = 81\) is not the largest square smaller than \(126\); there are two others, \(10^2 = 100\) and \(11^2 = 121\), leading to decompositions \(126 = 10^2 + 5^2 +1^2\) and \(126 = 11^2 + 2^2 +1^2\), and to corresponding decompositions of \(2016\).
  • I stopped looking for square factors too early, missing \(126 = 9 \times 14\) with \(14 = 3^2 + 2^2 +1\), instantly yielding the decomposition \(2016 = 36^2 + 24^2 + 12^2\).
  • notice that we can make four non-zero square instead of three by observing that \(10^2 = 6^2 + 8^2\) and \(126 = 10^2 + 5^2 +1^2 = 8^2 + 6^2 +5^2 +1^2 \).

What is the moral of that story? It illustrates something that Tony Gardiner calls structural arithmetic, see his paper Teaching mathematics  at secondary level. This is Key Stage 3 and 4 material, and, in mathematics learning, could be  an excellent preparation to elementary algebra.  As said earlier, mental arithmetic is not optimal way of solving arithmetic problems, but structural arithmetic, with pencil and paper, is.


Stalin on Mathematics

A paper


contains a tiny, but exceptionally important piece of evidence of Stalin’s attitude to mathematics:

Текст доклада Лысенко первоначально состоял из 10 разделов и занимал 49 страниц. Сталин зачеркнул весь второй раздел доклада, который имел название “Основы буржуазной биологии ложны”, сохранив в нем только абзац с критикой физика Э.Шредингера и написав против него на полях: “ЭТО В ДРУГОМ МЕСТЕ”.37 [Примечание 37: Прописными буквами дан текст, написанный Сталиным собственноручно на первом варианте доклада Лысенко.] В этом же разделе Сталиным было подчеркнуто положение: “Любая наука — классовая” — и на полях написано: “ХА-ХА-ХА… А МАТЕМАТИКА? А ДАРВИНИЗМ?”.

So, Stalin did not believe into the class nature of mathematics. This had profound impact on the fate of mathematics in the Soviet Union. Not every direction  in science was so lucky.

Full text of paper by Esakov

Source: В.Д.Есаков. Новое о сессии ВАСХНИЛ 1948 года //
Репрссированная наука, вып.II, СПб.: Наука, 1994, с.57-75.


The only way to learn mathematics …

It is a well known dictum:

The only way to learn mathematics is to do mathematics.

I would make it a bit more sharp:

The only way to learn maths is to use it for learning more advanced maths.

This is because to learn mathematics, we have to use it, and the is no more challenging and more stimulating use of mathematics than learning the next level, more advanced

As a corollary we get The Law of Supernumerary  Learning of Mathematics:

To be able to use mathematics at a certain level you have to learn it at the next level.

The law has an obvious empirical confirmation: graduates of mathematics departments of British Universities (BSc or MMath Degree) are hired by banks and insurance companies to do office jobs that require high school level mathematical skills. When big employers really need university level mathematical skills, they advertise for people with MSc or PhD in mathematically intensive disciplines.

In respect of teaching of mathematics, The Law of Supernumerary Learning of Mathematics implies that middle- and high school mathematics teachers should have an university degree in mathematics or mathematically intensive discipline, and primary school teachers
— at least good middle school level (such as GCSE) or high school level education in mathematics.


Some motivational slogans

From a  colleague’s letter:

Here the teaching has finished and the exam period has started. I had some midterms couple of weeks ago in which a student was trying to prove \(\sqrt{2}\) is irrational. they defined a rational number \(\frac{a}{b}\) with \(b=0\).

I think at that point mathematics upped and left the country. I had another who had made some calculation mistake in the first step of a proof by induction, and ended up with \(8>9\). They dutifully then marked it as \(P(1)\) holds and continued.

I had others that started with \(8 \mid 5^{2n} -1\)  (\(n>1\) ) and put it “equal” to many things and ended up with
\dots = \frac{8}{5^{2n} -1 } = 3
At this point I think logic decided to follow maths out of the country …

What can I say?

Keep calm and carry on.

Wikipedia says about the now famous poster:

“Printing began on 23 August 1939, the day that Nazi Germany and the USSR signed the Molotov–Ribbentrop Pact, and the posters were ready to be placed up within 24 hours of the outbreak of war. Almost 2,500,000 copies of Keep Calm and Carry On were printed between 23 August 1939 and 3 September 1939 but the poster was not sanctioned for immediate public display. “

Apparently the Blitz was not judged to be desperate enough situation — I personally see a great moral lesson in that, especially for us, mathematics teachers.


A response to “Malta: new “Learning Outcomes Framework””

Thank you Alexandre for taking an interest in the curriculum being developed for the Maltese schools. (As a matter of information, this curriculum is being developed by a consortium of foreign “experts” supported by a European Social Fund grant. What is shown on the website is work-in-progress, and one hopes that the final product will be a more coherent curriculum and banalities like the one you pointed out will have been removed.)
So, let me share my answers to the same question you ask, basically why does this draft curriculum contain such a statement: I can use equivalent fractions to discuss issues of equality e.g. gender. I agree with your two responses, namely mis-use of vocabulary and the strictures imposed by an Outcomes Based (OB) curriculum. But allow me to elaborate further.
In my view, the above statement would be banal whether one uses the term “equivalent fractions” or “similar fractions” or any other notion which extrapolates from 1/2=2/4=3/6=etc to anything having to do with gender equality. The problem, in my opinion, is that some people do not realise that, in science, we expropriate a word from everyday vocabulary to use in a context which does have some similarity to the everyday use of the word, but whose meaning becomes something technical which cannot be exported back to the everyday sense of the word.  I sometimes taught classes of Arts students who felt they needed to use some mathematical jargon in their essays (a few years ago the fashionable thing to do was to drop the words “chaos” and “fractal”). One of my usual examples of how wrong this is involved the use of the word “work”, as used in science and in everyday life. Translated into the context of curricula, the analogous banal statement could be something like: I can calculate the work done by a given force moving an object through a given distance and I can use this to discuss the conditions of work in factories and industry. 
What surprises me when statements such as the one on gender equality are made is that while the ambiguity of language is appreciated outside science, in fact it can be a wonderful tool in the hands of a good writer, when transporting scientific vocabulary back into the everyday world, this variegated meaning of the same word in different contexts is sometimes forgotten. I have no explanation why this happens.
But another problem with curricula written in OB style and which could have a bearing on such wording is the necessity that the statements should be written in a way that the learning child would write them, for example, by starting the description of each outcome with “I can…” That sentences such as the one you quote about gender issues crop up is not, in itself the main problem, in my opinion. Such sentences can be edited out when reviewing the curriculum. The problem, as I see it, is that this style excludes the possibility that the curriculum contain concepts to guide the teacher but which the student would not likely be able to express. So take your improved statement of how mathematics can help understand social inequalities:
I believe in the power of mathematics and I am convinced  that comparing numbers (for example, salary)  reveals a lot about gender inequality (and other, frequently hidden,  inequalities in the world — just recall the Oaxaca Decomposition and its role in fight against discrimination of any kind).
It might be reasonable to expect a Level 5 student (aged 7-8) to express such a statement up to “gender inequality”, but hardly the rest of the statement, although the writer of the curriculum might very well want to make a reference to the Oaxaca Decomposition to give the teacher an example of a highly non-trivial use of mathematics in this context.
This OB format, I believe, betrays a fallacy about the teaching of mathematics, namely that teaching elementary mathematics to 7-year olds, say, does not involve deep knowledge of mathematics, certainly not deeper than what a 7-year old can express.
I look forward to reading other comments, especially by readers of this blog who are more familiar with OB curricula than I am.

A few similes about mathematics

Let us first compare mathematics with art. As the historian and philosopher R. G. Collingwood said in his The Principles of Art, 1938

[it is not true that] …  ‘all art is quite useless’, for it is not; a work of art may very well amuse, instruct, puzzle, exhort, without ceasing to be art, and in this ways it may be very useful indeed.

[but] …

what makes it art is not the same as what makes it useful.

Similarly,  mathematics is useful but what makes it mathematics is not the same as what makes it useful.

Mathematics has its own intrinsic needs that have to be addressed for it to stay alive.

Now let us compare mathematics with something unambigously useful: a cow. The cow is useful indeed, it gives us cream (single, doubled, clotted), butter, cheese, milk and skimmed milk — the list can be continued.

Applied mathematics can be compared with the cow’s udder — it produces milk.

Some branches of pure mathematics are best described as the cow’s immune system — they keep the cow alive.

The cow of course has other uses. To make a steak, it suffices to take a piece of cow and gently fry it to taste. What is a piece of a cow? A mathematician.

Financial industry, security sector, etc. are connoisseurs of a good steak. NSA advertises itself as the biggest employer of mathematicians in the USA.

Some people claim that pure mathematicians’ focus on “useless” artificial problems “they invent for themselves”. But let us look at geneticists’ obsession with a pretty useless creature: Drosophila melanogaster.

An article in Wiki devoted to it says:

The species is known generally as the common fruit fly or vinegar fly. Starting with Charles W. Woodworth’s proposal of the use of this species as a model organism, D. melanogaster continues to be widely used for biological research in studies of genetics, physiology, microbial pathogenesis and life history evolution. It is typically used because it is an animal species that is easy to care for, has four pairs of chromosomes, breed quickly, and lays many eggs”.

Very frequently, “famous” mathematical problems are means of concentrating effort of generations of mathematicians on development of methods of proof in particular areas of mathematics, they are drosophilas of mathematics. The Last Fermat Theorem is perhaps the most famous example. In some cases (and the Riemann Hypothesis is the archetypal case) they, however, have exceptional importance for mathematics as a whole.

Дети, любите корову – источник мяса!

[Children, love the cow – a source of meat!]


Why were soviet mathematics/physics textbooks so insanely hardcore in comparison to US textbooks?

The most complete answer from a discussion in Quora:

Alex Sergeev, PhD in Physics

As I understand from reading comments, the OP means not school textbooks, but university textbooks, in particular Landau-Lifshitz was mentioned. In such case, I have to disagree with most answers presented.

Firstly, yes, they are indeed noticeably more hardcore than courses of a similar level in the US. Enough to compare two classic courses: Landau-Lifshitz and Feynman Lectures (which are, in turn, not really a walk in a park either, there are plenty of friendlier books). Same can be said about mathematical analysis books which I encountered. Soviet textbooks just go straight to the point and throw lots of definitions and formulas at you, without any preparation. The US textbooks try to explain simple things in more detail, and increase the complexity as they progress.

The reason for it, I think, is the difference in education systems. In the US, the point of education system is to teach students, as well as possible. In the USSR, the point was to get rid of weaker students and have only very good ones left, who would understand the subject no matter how hardcore the approach to it is. It might be more psychological rather than intentional, but in Soviet times it was a general sentiment: if you can’t do it straight-away, you are simply not good enough and should do something else. The US system tries to improve students and then select the best, the Soviet system tried to select the best and then improve them. The US system tries to make geniuses out of average students, the Soviet system tried to select geniuses disregarding average students. I might be a bit too categorical with this, but I don’t think it is too far from truth.

Another possible reason, stemming from the above is a lack of competition. In the US, the education system is adapting to students’ need, if the books are not teaching good enough they get replaced or amended. In the USSR, the textbooks were centrally selected and approved, and students had to adapt to whatever they were given.

Edit: I also have just recalled this phrase very widely circulated during Soviet times: “We don’t have irreplaceable people”. (It actually originated much earlier, and was used by Woodrow Wilson, but is widely assigned to Stalin, who in fact never said anything like that. I also believe that the connotation was intended to be different.) This phrase, however, well demonstrates the psychology of Soviet system. No one cared if you fail, there’ll be another person who’d take your place. In the US, if student is struggling, it is partially a teacher’s fault; in the USSR, it is 100% student’s fault.