Train Carriages

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
Age
5 to 11
Challenge level
filled star filled star empty star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



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.

Train Carriages



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

Train Carriages



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

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

That's all really.

Here are some to start you off.

Train Carriages



You could draw these.

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

Train Carriages


Next:-

Train Carriages



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

Train Carriages


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

Remember the four "Rules" above.

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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.

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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.