I told my husband that if we weren’t already married, I’d run away with nrich in a heartbeat.
That being said, it’s probably no surprise that my favorite problem comes from nrich: consecutive sums and is my response to the explore the mtbos mission 1.
I’ve used this problem a few times, with high level students, low level, and in between as well.
Here’s a poster with a general overview of the problem. The link from nrich provides starting help as well as teacher notes and a solution.
Some things I love about it:
1. So many points of entry and a low barrier.
2. So many paths. I’ve had so many different conjectures arise from this problem because of the open-ended nature of it and its ease of exploration. The numbers are not intimidating so students are unafraid to explore some of their findings.
3. Multiple extensions. For example, do consecutive differences work similarly? What about consecutive products? Or better yet, the difference of consecutive products!
4. Students organize their work in so many different ways. It’s completely fascinating to see it happen.
When doing this with a lower level class, I usually have them make a list of noticings and/or wonderings. This way, the patterns they learn to communicate what they believe to be true in their head. I may challenge them to generalize a little, especially with odd numbers always being a consecutive sum.
The most exciting thing that happens when I do this problem is the “what ifs” that students can’t help but think up themselves. For example, What if we took the difference of consecutive numbers? What if we took the sum of consecutive odd or even numbers? Consecutive square numbers? Triangle numbers? Negative numbers? It’s pretty amazing to be part of.
If you have tried or try this problem in the future, I’d love to collaborate on it.
Number Loving Beagle 
This sounds like a wonderful open ended question. I will have to try it with my eighth-graders. What I love most is that it is a open-ended question that requires deep thinking and yet like you said it has a low barrier so lots of children can be successful with it.
It looks like a great activity to present to students and ask them what they notice about the numbers. I might put this up and just ask them what they notice. Some great dialog and discussion is bound to happen. Then show them the video afterwords.
This is great – I haven’t tried it, but I am going to see where it “fits” in with what we’re doing. I see some similarities here and the task: Use the four operators + – * / and the number “4” four times to come up with all numbers 1 – 10.
What I love about questions like this one – but my students often do not – is the feeling of disconnectedness from the day to day work. Notice I said ‘feeling of disconnectedness’ not actual for real disconnectedness. Of course, any thinking along these lines enriches our other thinking processes as well. These kinds of problems introduce a sense of play that is crucial in the classroom.