Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
Investigate the numbers that come up on a die as you roll it in the
direction of north, south, east and west, without going over the
path it's already made.
Well, how about going a bit wild and designing a new shape for
these twenty eight squares in which the numbers follow on in some
way that is not too difficult to follow? For example, you could put
the twenty eight squares into a triangle and decide to go in an
up-and-down order like this:-
You can then go further by looking at the way the numbers are
arranged and see if you can spot any patterns. For example in the
first usual way:-
1. If you take any two by two square (like the one in at the top
left-hand corner with $1$, $2$, $8$ and $9$ in it), then the
opposite corners of that square add up to $10$. Then, if you take
the next two by two square along [with $3$, $4$, $10$ and $11$ in]
then the opposite corners add up to $14$ and so on.
2. Going down any column the numbers increase by $7$.
Do let us know what you discover in the designs you have
created. Can you explain why the patterns occur?
Then ask "I wonder what would happen if ...?''