My unsolved problem - the magic triangle.

The #mathscpdchat today about perseverance today I suggested an interesting point - share your unsolved problems with them. I was asked for more information and so dutifully supply it here; it is little more than a curiosity in reality but I have been puzzling it through for nearly a decade and have never managed to solve it to my own satisfaction. It centres around completing the magic triangle, like the example below.

For those who haven't come across it before, the point is this - you use the numbers one to nine to complete the three sides of the triangle so that they all have the same total.

Now solving these is not a problem (I probably wouldn't be much of a maths teacher if it were!) but my unsolved part is related to how many solutions. Let me explain in more detail:

It is relatively straightforward to prove that the 3 corner values have to add up to a multiple of 3 (although still a lovely example of a proof for GCSE pupils), but it is also relatively straight forward to find combinations of 3 numbers that do sum to multiples of 3, but cannot be used as the corners of a magic triangle (for example, you cannot complete a magic triangle with 3, 4, and 8 in the corners despite the fact they add to 15). My unsolved problem has been finding a sufficient condition as to which multiples of 3 work. Now I can find them by brute force, but I want an underlying property, I want to understand why certain combinations will and certain combinations won't.

For the longest time I thought it was only sums that lie on straight lines through this grid

                            

                                                 1                  2                    3

                                                 4                  5                    6

                                                 7                  8                    9

(which by the way, can all make magic triangles) but then when I had some Year 6s in a masterclass working on finding them, one of those pupils found a solution that doesn't lie on a straight line through the grid (which I honestly can't remember but which wasn't a problem as I couldn't prove why it would only be lines through the grid that would work in the corners anyway). I have tried rearranging the numbers into a magic square and looking for patterns in that arrangement, but to no avail.

I shared my struggle with my Year 7 pupils today, as we were talking about mathematicians and problem solving, and why not share with your pupils the mathematics that you have puzzled over?