3 Rings

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
Age
7 to 11
Challenge level
filled star empty star empty star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



I found a number of small bracelets and rings on a table some time ago and I noticed how some were on their own, others were touching at the edges, others were overlapping each other and some small ones had found themselves inside larger ones.
3 Rings
3 Rings
I took two of these, one ring and one bracelet, and explored what possibilities there were.

I thought that this would be the next challenge for you all. To look at the situation when you have three rings, circles, bracelets . . . . it doesn't matter what they are really or what size they are. They could even expand and get bigger or get smaller if you liked. But, thinking of the four things I noticed at the start:-

1) TOUCHING

2) OVERLAPPING

3) SEPARATE

4) IN/OUT SIDE

I wonder what would be the number of ways in which 3 such circles could be?

Here are some ways, remember I said they could be different sizes each time, but I've coloured them so that it is easy to know which one we are talking about.

3 Rings


Well I feel you could carry on at this point, just a few points to remember:-

When writing you must say something about each of the three circle/rings/bracelets.

Three separate ones could be anywhere yet separate and they would all count as one arrangement, and the same kind of things goes for any other arrangement, if the words are the same then, for this challenge the arrangement is the same.



You could now ask "I wonder what would happen if.....?"