This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?
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.