One of those classic investigations that gets forgotten about all too easily. So much scope for generalising at different levels.
The fact that all odd numbers can be expressed as sum of two consecutive numbers is probably the first thing that will be established. But why is this the case? And can students express this as a generalisation, first in the form of concise words and then algebraically?
The beauty of this is that there are then many other layers of things to discover, right down to a generalisation explaining which numbers cannot be expressed as a sum of consecutive numbers. And maybe even a proof.
This nRich page gives away some of the answers.
Thanks to Alan Parr for reminding me about it with this excellent blog post: