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## 'Train Carriages' printed from http://nrich.maths.org/

I was on a train the other day and I was looking out of the
window as we went through a station and I saw lots and lots of
carriages waiting in another line. They were going to be linked
together to make a new train going on a long journey - so I thought
!!

It was then that my train went round a curve in the track and I
looked out of the window and saw the front of my train and quickly
turned my head and saw the back of the train.

After a few more miles I had seen that my train had ten
carriages to make up the whole train.

I thought back about the carriages I had seen at the station and
wondered about making them into several trains.

I thought my train was rather special having ten carriages so I
want to put this challenge to you all.

- Suppose there are $24$ carriages.
- They're going to be put together to make up some trains.
- The smallest train you are allowed is one with
two carriages.
- You must include at least one "ten-carriage train".

That's all really.

Here are some to start you off.

You could draw these.

You could use cubes/ beads/ boxes or whatever to stand for the
carriages. So it might also look like:-

Next:-

and if you used squares or cubes it might look like:-

So now it's your turn. See what different train arrangements you
can make.

Remember the four "Rules" above.

It would be good to think if you have been using some kind of
special way of getting new ones. Maybe you've found a pattern that
helps you get lots of answers?

Please write to us and tell us about your ways of doing
this.

FINALLY

You simply have to ask "I WONDER WHAT WOULD HAPPEN IF ...?"

Here are some to start you off.

"I wonder what would happen if I had to only make
three trains?"

"I wonder what would happen if I had to make only
four trains?"

"I wonder what would happen if I had to have all the trains
different lengths?"

and so on

and so on

and so on.