In writing my new book 'Practising Maths' I referenced a lovely result (you will have to buy it to see how) that sums such as 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1, etc. all produce square numbers.
If you haven't come across this result before then feel free to have a look at it for a minute (even try and prove it) - if you are familiar with consecutive triangular numbers summing to square numbers, it is closely related.
The curiosity I noticed was that I knew 121 was also square. So I became interested in the fact that 1 + 2 + 1 is square, and 121 is square. I decided to look into the others, and it turns out they are also square! Well, the ones up to 12345678987654321 are square anyway.
...
This of course raised a question - is this a reflection of something deeper? You might like to spend some time exploring and coming to your own conclusion before you read on.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
I guess the truth is a little of both.
If we consider squaring polynomials of increasing order with unit coefficients we get the following:
These are, of course, the same expressions as above, but in base x rather than base 10. So, if we substitute x = 1 into the expressions we get the sums on the left of the above table. However, if we substitute x = 10 into the same expressions, we get the numbers on the right of the table.
In terms of a task, we could offer the first few rows of the table to pupils and ask them what they notice/wonder. They might explore when the pattern breaks and why. We might encourage them to write out the numbers using explicit base 10 notation, such as 1 × 100 + 2 × 10 + 1 and see what insights this brings out. Pupils with the necessary algebra skills might even explore the expansions given above. Or we might just show it to pupils as an example of a mathematical curiosity.