Enough people have noticed the Consecutive Numbers piece to justify a follow-up.
Discovering just which numbers can’t be made by summing a set of consecutive positive whole numbers is such an elegant and surprising result it brings a smile to the face. On the way – and worthy of being an enquiry in its own right – is the key observation that while any set of three consecutive numbers sums to a multiple of 3, four consecutive numbers do not sum to a multiple of 4.
There are other spinoffs as well.
* Those sets which start with 1, e.g. 1, 2, 3, 4 sum to give the triangular numbers.
* What if you do the Consecutive Numbers enquiry with just the consecutive odd numbers? In this case, of course, those sets which start with 1 (e.g. 1, 3, 5) sum to give the square numbers.
* Which rather suggests that if you use numbers from the series 1, 4, 7, 10, 13, … you ought to find something interesting.
And that leads me to Roof Numbers, which you probably won’t have heard of. I was asked to work with some B.Ed. students, and in every respect except one I was given a totally free hand. So I was able to create a course built around exploratory maths, and a quite wonderful term it was too (I was asked to give them a second course the following year, and they insisted I was a guest to their graduation).
The fly in the ointment was the university’s requirement that I set a timed unseen written examination. But at least I got to set it, and Roof Numbers were my response. If the Consecutive Numbers question is a highly open problem, then Roof Numbers are hyper-open.
Here’s the problem:
Start with a bottom row of dots.
Above it, add a row which is three shorter than the bottom row.
Keep going till you feel like stopping, or until it’s impossible to carry on.
You have made a Roof Number.
o o o o o
o o o o o o o o
o o o o o o o o o o o
So 24 is a (level 3) Roof Number.
11 is a level 2 Roof Number:
o o o o
o o o o o o o
What can you find out about Roof Numbers?
I promised anyone who was desperate could buy a hint, but I knew perfectly well that anyone who’d spent a term doing problem-solving investigative mathematics would be able to spend their hour finding out interesting things, such as:
* are there numbers which are roof numbers in more than one way?
* are there numbers which cannot be made as roof numbers?
* what can you find out about level 3 (for example) roof numbers?
And since the step size of 3 is wholly arbitrary, you could just as well have roof numbers where the step size is 4, or 2 – and if the step size is 1, then you get the original Consecutive Numbers enquiry as simply a special example of Roof Numbers.