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Joe and Sarah described their solutions very well, although they didn't find all the possible lines of four. Here is Joe's:
Hannah wrote to us to tell us about her work - she is still finding possible solutions but says:
Clement Chan from Sha Tin College in Hong Kong has found thirteen ways. He says:
I first tried to find all the possible lines I could make on the grid than looked at each one carefully for example:
I want to make a vertical line from the $1$, I check whether the $21$ can go in anywhere and if it can I check whether the $3$ can go anywhere, and then the $2$ and put the $1$ on to finish the line.
If I want to make a line diagonally from the $5$ then I check it in the same way, and if I want to make a horizontal line from the $15$ I check it the same way etc.
Laquiesha and Chloe from Kingsmoor Primary also thought they had found thirteen ways, but they didn't list them all. They pointed out:
We also found out that you couldn't do diagonals because the no. $21$ had to be placed on the four odd numbers $1$, $3$, $7$, $21$ and you couldn't put the number $2$ in the diagonal because $2$ needs to be on a even number but they were all odd in the diagonal.
(In fact, $2$ could go on the $1$ in the diagonal, couldn't it? But I see what you mean, that you can't then place the other two numbers in the diagonal.) Ben T and Ben H from Brewood Middle School realised that diagonals are impossible too.
Mrs Rankin from Sandown Primary School wrote to say:
Well done all of you - you're right. We thought that there were only fourteen solutions. We obviously weren't being as systematic as we thought! Here is a table which shows the group's solutions:
I like the way you have organised your recording. I have highlighted the two $21$s in the bottom of your table because I think in fact these should be $7$s. I hope you agree with me.