Here we’ll look a bit closer at square and triangular numbers, but I’ll offer a bit of insight into polygonal numbers in general.
Polygonal numbers are numbers that can be represented as points on a plane arranged into a polygon (and they fall under a broader umbrella known as figurate numbers. The idea dates back to ancient Greece, where lots of mathsy things were done with pebbles and sand.
You’ll be familiar with triangle numbers:
and square numbers :
Both have an nth term. Square numbers (also known as “perfect squares”) have the formula n2, and triangular numbers have the formula n(n+1)/2. Both can be derived using a simple sequence table, and it’d be worth doing so with your students. Obviously we’re talking quadratic sequences, not linear.
The term ‘squared’ is in fact slightly different to a square number. Squaring a number is the application of the formula n x n and is perhaps best thought of alongside cubed etc rather than polygonal numbers, whereas a “square number” is a number that can form a square using units. For clarity, consider a triangle number. I don’t say ‘4 triangled’ when I apply n(n+1)/2. Furthermore, (2.4)2 is not a ‘square number’.
Anyway, each polygonal number has it’s own strict formula that can be derived by school students if you so feel the need.
Now, there are some interesting relationships between square numbers, and between triangle numbers, and between square AND triangle numbers.
Let’s talk about Gauss.
You’re possibly familiar with the age-old tale of Gauss being asked to sum the numbers 1-100, and he did it super quick with no working out (smart git). He did it by figuring out the sum of 1-100 is 1 + 100, 2+ 99, 3 + 98 etc rather than just mindlessly adding them sequentially. If you hadn’t noticed, that’s just 50 x 101 = 5050
Now, what’s the 100th triangle number? 100(100+1)/2 = 5050. Coincidence? No sir.
Let’s take the sum of all numbers up to 6. You can work that out in your head as 6+5+4+3+2+1 = 3 x 7 (Gauss) = 21.
6(6+1)/2 = 21. Let’s visualise what’s going on:
So essentially, the sum of all consecutive integers ‘n’ *is* the triangle number ‘n’.
Fun huh? Similarly, the sum of consecutive odd numbers is a square number.
So if I said ‘sum the first eight odd numbers together’ i’d just do 8 x 8.
Or if I said ‘sum the odd numbers from 1 – 50’ I’d just do (50 / 2)2
(there are 50/2 odd numbers between 1 and 50)
And with some jiggery pokery you could answer ‘sum the odd numbers between 120 and 210’ but I’ll let you ponder that one on your own.
So there are lots of fun patterns using square and triangle numbers. That’s just a couple of them.
Finally we can also prove visually that the sum of two consecutive triangle numbers is always a square number:
So whenever you’re talking about triangle numbers, be sure to stack them in a right-angled triangle rather than a kind of pyramid. It makes for easier proof visualisations. I wonder if the new exam emphasis on proofs will allow for visual proofs? Hmm…
Random aside: there is actually a different ‘type’ of square and triangle number which has a similar name but different properties. The ‘centered square‘ numbers, and ‘centered triangle‘ numbers look like these:
Different patterns, different numbers involved, and different formulae. Pretty.
Now, before we finish, you may have been wondering about the term ‘polygonal numbers’. It alludes to OTHER shape-based numbers. And there are many other types. Infinite in theory. They get a bit boring and nameless later on, but here are a few:
And they all have ‘centered’ versions too, which are prettier:
(centered nonagonal numbers)
Eventually things get dull and we call them ‘23-gonal‘ etc. 1166 is a 23-gonal number, but who cares.
Some numbers are extra special, and they are more than one shape! 36 is both square and triangular for example. I find that strangely exciting. As do many geeky maths people. There are a few mysteries attached to multi-polygonal numbers (I made that term up, at least I thought I did. Turns out it’s in use.) 9801 for example is both square and pentagonal (apparently). But is there a number that is triangular, square AND pentagonal? These are the nerdy things mathematicians try to discover. (hint: no-one has found one yet).
I’ll leave you with this little puzzle:
Arrange the numbers from 1 – 16 in such a way that each pair sums to equal a square number. (Taken from this amazing book by Matt Parker)
(that’s a flippin’ cube, not a square).