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.