Three block towers
Problem
Take three different colour blocks, maybe red, yellow and blue.
Make a tower using one of each colour.
Here's one with red on top, blue in the middle and yellow on the bottom.
Now make another tower with a different colour on top.
How many different towers can you make?
How will you know that you have found them all?
You can click below to see how two learners started this task.
Elias says:
Then I moved red to the bottom and moved yellow and blue up to make a new tower:
I did the same again, moving yellow to the bottom:
I did this again, but realised that would give the first tower. So I went back to my first tower and kept the bottom cube the same, but swapped the top and middle.
Did you start the problem in the same way as either of these children?
What do you think about each method?
You may like to print off one of these sheets for recording three-block towers.
Click here for a poster of this problem.
Printable NRICH Roadshow resource.
Getting Started
How many towers will have a red block at the top?
How many will have yellow at the top? What about blue?
How will you know when you have got them all? Perhaps you could rearrange them to help?
Student Solutions
These solutions (from a range of countries!') demonstrate some good thinking.
William from Beverly Farms Elementary in the U.S.A. wrote:
First I put red on top and did red white green and then I did red green white.
Next, I did green red white and then I did green white red.
Finally, I did white red green and then white green red.
Manisha from the Dehradun Public School in India sent in the following:
In 3 Block Towers puzzle, I first kept any one colour at the bottom and remaining at the 2nd and 3rd position that made one tower then I changed the 2nd and 3rd block means 2nd at 3rd position (top) and 3rd at the 2nd position that became the 2nd tower.
In this way, now I have put other colour block at the bottom that was the 3rd tower and changed the position of the 2nd and 3rd block so got the 4th tower.
In this way, now I have put the remaining colour block at the bottom that was the 5th tower and changed the position of the 2nd and 3rd block so got the 6th tower.
In this way, we can make a total of 6 towers.
Phoebe and Cai from Ysgol Porth y Felin in Wales had their work sent in:
Phoebe chose to work out her problem by colouring the squares in her book with the options.
On the second part of the challenge, Cai realised he didn't have to draw each option in his book, once he figured out green on top had 6 different options, he used his times tables to work out the possibilities.
(Click on these images and a larger version will open in a new page.)
Lucy from Hawkesbury Primary wrote:
Lucy solved this by trial and error, quickly spotting that after finding three possible options, that red had been in each place once, but the other colours hadn't yet been in each place, so there must be more solutions.
Sergio from El Centro Ingles in Spain sent in the following:
I did first with red at the top and this is how you can do it: (red, yellow and blue) and (red, blue, yellow).
When we finish with red at the top you can do yellow or blue. I am going to do the yellow and these are the orders: (yellow.red.blue) and (yellow, blue, red)
And for finish we have the blue at the top: (blue, yellow, red) and (blue.red.yellow)
That is how you do this.
Teachers' Resources
3 Blocks Towers
Take three different colour blocks, maybe red, yellow and blue.
Make a tower using one of each colour.
Here's one with red on top, blue in the middle and yellow on the bottom.
Now make another tower with a different colour on top.
How many different towers can you make?
How will you know that you have found them all?
You can click below to see how two learners started this task.
Elias says:
Then I moved red to the bottom and moved yellow and blue up to make a new tower:
I did the same again, moving yellow to the bottom:
I did this again, but realised that would give the first tower. So I went back to my first tower and kept the bottom cube the same, but swapped the top and middle.
Did you start the problem in the same way as either of these children?
What do you think about each method?
You may like to print off one of these sheets for recording three-block towers.
Click here for a poster of this problem.
Printable NRICH Roadshow resource.
Why do this problem?
By providing opportunities for your class to reflect on different ways of solving a problem and talking about them, learners can begin to appreciate the problem-solving journey and not just the answer.
Possible approach
Start with a whole class activity in which you invite a child to make a three-block tower for all to see. Then ask someone else to make a different three-block tower. Invite other leaners to explain why the two towers are different. Repeat the process so that another different tower is made, again asking someone to explain why this third one is different from the first two. At this point, introduce the task and give learners time to work in pairs. You may like to give out this sheet to aid recording.
Once everyone has had chance to create a number of solutions, but not to have completed the task, share some different ways of approaching it. You can do this by drawing on methods that your learners have come up with, or by using the examples of Jemima and Elias in the problem. (If the latter, this sheet of the problem and two methods my be useful.) Invite everyone to look at each method and understand it, before facilitating a general discussion about the two different approaches. Did any pair use one of these methods? What do they like about each? What do they not like? Give time for pairs to continue to work on the problem, but invite them to choose one of the approaches they have heard about, if they so wish.
In the plenary, you could ask a couple of pairs to explain why they changed their approach, or not. Allow time to then ask one pair to try to convince everyone that they have found all the possible towers. (You may want to have warned a few pairs beforehand so they have chance to refine their arguments.)
Key questions
How can we be sure we have the different solutions?