The tall tower
Problem
The Tall Tower printable sheet
You have been imprisoned at the top of a tall tower by a wicked magician!
You can get out by climbing down the ladders. As you come down you collect useful spells.
You can go down the ladders and through the doorways into an adjoining room, but you cannot go into the same room twice, nor climb up the ladders.
The numbers in the rooms show how many spells there are in each one.
Which way should you go to collect the most spells?
And which way to collect as few as possible?
Can you find a route that collects exactly 35 spells?
By clicking below, you can read how some other children started this problem.
Krishan says:
I thought that if I want to get the highest number of spells I need to visit as many rooms as possible.
Hiromi says:
I wondered whether it was possible to visit all of the rooms in counting order.
Fay says:
I chose a route and then added up how many spells I had collected.
Did you start the problem in the same way as any of these children?
What do you think about each method?
Getting Started
Which might be the best number to leave out?
Which is the shortest route?
Might it help to record the routes you take so that you do not repeat yourself?
Student Solutions
These solutions show some interesting ideas, thank you for sending them in.
Gabby from Walthamstow Hall Junior School said:
I thought that to get the highest number of spells in total it would not matter if I didn't go into as many rooms as I could, it would matter that I could get into most of the rooms with higher numbers of spells in it so I could get a higher total. I then tried a couple of solutions to see which route would have the highest total of spells.
Casey from a school in Australia wrote:
You go through room 1 to room 3 then 4 then 6 then 5 then 7 and then 9 that gets you 35 spells.
Which way should you go to collect the most spells?
A solution is room 1 then room 3 then room 4 then room 6 then room 5 then room 7 then room 8 then room 10 then room 9. So then that gets you 53 spells.
Igor from the International School of Luxembourg sent in the following:
On the left side there are all the odd numbers and on the right, there are the even ones. In this example all the even numbers on each floor are bigger than the odd ones so to collect fewer spells you go left and to collect more you go right.
I'm almost 100% sure that there is only one way of doing the puzzle but I don't think I could come up with it.
Jacob from Moulton CEVC Primary School wrote:
To collect exactly 35 spells I started with subtracting the number you have to go through to exit the tower (9) then I worked out how to make 26 with the numbers left.
Teachers' Resources
The Tall Tower
The Tall Tower printable sheet
You have been imprisoned at the top of a tall tower by a wicked magician!
You can get out by climbing down the ladders. As you come down you collect useful spells.
You can go down the ladders and through the doorways into an adjoining room, but you cannot go into the same room twice, nor climb up the ladders.
The numbers in the rooms show how many spells there are in each one.
Which way should you go to collect the most spells?
And which way to collect as few as possible?
Can you find a route that collects exactly 35 spells?
By clicking below, you can read how some other children started this problem.
Krishan says:
I thought that if I want to get the highest number of spells I need to visit as many rooms as possible.
Hiromi says:
I wondered whether it was possible to visit all of the rooms in counting order.
Fay says:
I chose a route and then added up how many spells I had collected.
Did you start the problem in the same way as any of these children?
What do you think about each method?
Why do this problem?
This problem allows children to practise addition and subtraction, and compare numbers, in an interesting and challenging context. It also provides an opportunity to encourage learners to reason mathematically as they justify their solutions.
By offering three different ways into the problem, you can capture pupils' curiosity. By focusing on different approaches to a task, learners' attention is on the mathematical journey rather than just the answer.
Possible approach
Show the image of the tall tower for the whole group to see and tell the 'story'. To ensure that everyone has understood the constraints of the task, take some time to draw a few routes on the board and to find the total number of spells collected in each case. It would be useful to draw at least one route which is forbidden by the 'rules' so as to provide an opportunity for
clarification.
Set pairs off on finding the route which collects the most spells (a sheet of towers might be useful for recording) but, if possible, do not give them time to find the solution. Instead, offer the three starting points from the problem, characterised by Krishan's, Hiromi's and Fay's methods, and
ask learners to try to understand them. You may wish to give children the printable sheet from the top of the problem page, which contains all of this information.
After a suitable length of time, bring everyone together again to facilitate a discussion about possible ways of starting this problem. Did any pair use one of the approaches on the sheet? What do they like about each one? What are they less keen on? Why?
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, and/or you could share ways of working on the two other parts of the task.
Key questions
Possible extension
Children could find how many different ways there are to go down the tower. Is it possible to collect all the different numbers of spells between the highest and lowest? Alternatively, learners could create their own version of the task by adding numbers of their choosing to some blank images of the tower.
Possible support
If arithmetic is a barrier to tackling this task, counters could be used to represent the spells on this large picture of the tower.