Change Around

I ran a session at BCME 9 entitled 'The NRICH Roadshow: Is it Just a Bit of Fun?'

At past conferences I have taken a bunch of Roadshow activities and given delegates the opportunity to explore whatever took their fancy but for various reasons, including packing space, I decided to focus my session around one type of equipment. I chose counters as they are cheap, easy to acquire, generic and not topic specific. I picked out a selection of activities using counters from different areas of maths and some which are suitable for a range of ages.

To start I introduced one activity for everyone to work on, Change Around. I had found it while I had been working on updating some of the interactivities and thought it'd make a nice intro activity.


Move just three of the circles so that the triangle faces in the opposite direction.

We can of course extend this to any sized triangle.

What is the minimum number of counters you  need to move to flip any sized triangle upside down?

The beauty of this problem is that the sequence isn't a standard sequence from the English curriculum and convincing yourself that you've found the minimum number of counters isn't trivial. Have a go yourself if you have a few moments!

(If you haven't any counters handy you might like to use the interactive here. You can change the grid to a 'brick bond' style to help you arrange your counters triangularly.)

The thing that fascinates me about working with groups of teachers and students on non routine activities is the many ways of recording thoughts, ideas and results. I know I have my own way of seeing things but often looking at other approaches gives insights into the problem that I hadn't seen or highlights strategies that might be useful in future problems. Here are a few jottings that I noticed during my session:

My favourite is the bottom right. Mostly because it is completely not the way I think about this problem. I tend to love a diagram I often use a diagram even when it isn't necessary! But here, the strategy used to prove that a minimum had been found was pretty neat. 

The first column is the original triangle. then the next column is the number of counters needed to add to each row to flip the triangle if the top row remains as the top row. Then the next column is if the second row in the triangle becomes the top row. 

It took me a while to figure this strategy out and I did feel the need to make a spreadsheet at this point to check out the minimising...

For clarity, here is a diagram showing the second way. The original triangle is in red and blue, the red counters are removed and the green counters are added to create a flipped triangle.

I was intrigued by the 2 minimal solutions. but I think they are rotationally similar.

I enjoyed this session thoroughly and the discussions that this activity threw up were wide ranging. I find that when I am presenting a conference session it is most rewarding when I give the delegates the opportunity and space to share their ideas and I am able to learn from them too.