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'Castles in the Middle' printed from http://nrich.maths.org/
This is one of a series of problems
designed to develop learners' team working skills. Other tasks in
the series can be found by going to this article
What are you aiming to do?
For the task:
The team has to recreate the 3-D solid or "castle" (created by the
designer) by asking as few questions as possible.
As a team:
- Listening and finding out what others think
- Giving reasons for ideas
- Pulling ideas together
- Finding out whether the group is ready to make a decision.
You will be working in a team of four or five. You will need
two sets of cubes (multilink or similar). The image above uses nine
cubes of four different colours. You might wish to start with a
fixed number of cubes each of a different colour and then work with
cubes of the same colour later.
Choose someone in the group to be the designer. If you try
this task more than once, team members should take turns at playing
Give one set of cubes to the designer and the other identical
set to the rest of the team.
You may also wish to choose someone to be the observer (a
really good idea if there are more than four people in the
Tackling the problem
Without the rest of the group seeing, the designer creates a
castle using some or all of the cubes available.
Using the rules for asking questions, and checking that they
all agree first, the team takes turns to ask the designer questions
that will help them recreate the castle.
When the team thinks they have the same castle as the designer
they can check and the task ends.
At the end, the observer gives feedback about the way the team
worked together, highlighting strengths and ways that they could
improve next time. The team discuss the feedback and how they think
You are trying to recreate the castle
using as few questions as possible.
- You must agree any question before asking the designer.
- The team can only ask questions of the type listed in the
- The team must take turns in asking questions.
- The designer can only answer "Yes", "No" or give a number.
- You do not have to use all the cubes.
- The cubes should be arranged so that each cube has at least one
full face touching another cube and all faces must fit together
exactly (no overlaps).
- You only answer a question if the team has discussed and agreed
- You only answer questions of the agreed type.
- You can only say "Yes" or "No", or give a number as an
Team question rules:
The team can ask questions about:
- the number of cubes,
- the number of cubes of each colour,
- the number of cubes in each row, column or layer,
- the number of cubes touching a given cube,
- the number of cubes of different colours touching a given
- the colour of a cube in a particular position,
- the symmetry properties of the castle,
- any holes or gaps in the castle.
Your role is to:
- identify the times when reasons for ideas are given by members
of the team,
- check that everyone agrees before a question is asked,
- check that the questions fit those allowed in the rules,
- check that members of the team take it in turns to ask
- count the number of questions the team ask.