The Tall Tower

'The Tall Tower' printed from https://nrich.maths.org/

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The Tall Tower

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.

A tower of five floors, each with two rooms. The rooms are numbered 1 to 10 from left to right and then from top to bottom. The start takes you to Room 1, and the finish is out of Room 9 (both on the left). Each room has a door to the adjoining room and stairs to the room below.

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

Can you go through all the numbers?

Which is the best number or numbers to leave out? Why?

How any spells do you collect going that way?

How will you make sure you remember which routes you have tried?

Can you find a route that collects more/fewer spells?

How will you check your solutions?

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.