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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.

Multilink Cubes

If you had 36 cubes, what different cuboids could you make?


Age 7 to 11
Challenge Level


Sometimes in schools, churches, or clubs people like to make a small magazine. Perhaps you belong to a club or a group, and if you can use the photocopier it's maybe cheaper than using the computer printer.

So let's suppose that you are going to have 16 pages of A5 size, which you can get by using both sides of A3, [which is twice as big as your usual A4] and folding it in half three times. That is halve it, halve it again and finally halve that again.

I've discovered so far four ways of doing this but I guess there are more.


I've used small dotted lines to show 'valley' folds and large dashed lines to show 'mountain' folds.

Number 1 came by doing:- Right over Left; Bottom over Top; Left over Right.
Number 2 came by doing:- Right over Left; Right over Left; Bottom over Top.
Number 3 came by doing:- Right over Left; Bottom over Top; Right under Left.
Number 4 came by doing:- Right over Left; Right over Left; Bottom under Top.

Try these for yourself and see what others you can come up with.

Well that may be enough of an investigation for you and you might like to think about what happens when you do only two halvings [for an 8-page magazine] or four halvings [for a 32-page magazine]!

But those of you who wish to stick to the A3 [being halved three times] and 16 pages being stuck to both sides of the A3 ready for photocopying, let's have a look at the numbering of the pages. So that when you cut up all the folds except the centre one you are ready for stapling.

Here's what I found I had to do to get the numbering correct. Number 1 shows the front and then number 2 shows it turned over.

Try and number the pages for one of your foldings that was different from mine and see what you get. It's probably different, I would think it would be. I'm not quite sure about the stapling either?

These 16 numbers looked interesting so I've separated them from the drawing.

9 8 5 12  
16 1 4 13
11 6 7 10
14 3 2 15

I put the 'reverse side' numbers underneath.

Whoa! Left-hand pairs make 17 and so do the right-hand pairs!

I expect there's a lot more to see here.

Well have a go, and don't forget to say every now and again, "I wonder what would happen if I ...?''.

Why do this problem?

Like so many investigations, this one is like a journey through a wood, where we encourage children to keep all their senses alert. As I've suggested in the introduction to this challenge there are a number of things you can explore. The activity could be done by those who have a good enquiring mind using a trial and improvement method. It has spatial and numerical possibilities and it would be very valuable for you to tease out from children the ways in which they imaged the whole process.

Possible approach

I think that presenting it as written, talking it through with learners, will be adequate and there should be opportunities for the pupils to ask further questions before trying to form solutions.

Key questions

Are you happy with the way that it folds?
How are you making sure the numbering is ok?

Possible extension

If the solutions have been recorded in some kind of 4 by 4 table - as one is shown in the problem - then the properties of that table can be explored.

For more extension work

These pupils can look at the different ways in which the orginal large shee can be folded in order to give the $16$ pages. If solutions are found for each and every way of folding then the arrangement of the numbered pages onthe $A3$ sheet will need to be explored and relationships notes. Further work can then be done on producing 32 pages from the original sheet and the investigation followed in the same way.

Possible support

Some pupils will need help with getting into the physical understanding of the problem - for these pupils an already folded sheet may come in useful.