Year 8: Algebra - arithmagons

A lesson start (taken from a write up in ACs' 2nd year of teaching!)
The students filed in as usual, in a noisyish way, and slowly got their stuff out of their bags. I took the register over murmurings of conversations - choosing to ignore them. I then went to the board, with most eyes on me. I tapped with my pen and waited for everyone's attention.

We were going to do some work on arithmagons - I drew one in silence, put a number in two circles, paused and filled the box inbetween, paused...put a number in the third circle...filled in the other two boxes.

A few hands had gone up, there was silence. Another example, still silence, a couple of students bursting to tell the answers in the boxes, but still silence (this was surprising)...I was making eye contact with many of the class and looking a lot at a girl who I felt might be the last to pick up what was going on - there was concentration, but still no understanding on her face. A third example...I turned to look at the class and everyone's eyes were burning into the board - I hadn't experienced this before. Still silence...I now took two answers from the class for what should go in the boxes, everyone's hand seemed to be up except the girl; she was straining and seemed to have just twigged; she half whispered an answer, not quite committing herself - but it was correct. One more example...two boys had lost concentration, staring again brought them back. The girl's hand was now up with the rest - the boys seemed to be following, after all, so I nodded at her and a correct answer came.

I then drew an arithmagon with only the boxes filled in and invited the class to try to find what the numbers in the circles could have been...no one needed a further explanation, which, for me, is rare!

Questions:
· Can you find a method or strategy for solving any arithmagon?
· Are there any arithmagons that have no solution? (You might want to write on the board any the class come up with which they can't solve.)
· Is there ever more than one solution?
· Can you use algebra to find a solution? (or just to describe how they work eg if a, b and c are in the circles, what is in the boxes?)
· What happens with square arithmagons? pentagonal arithmagons?