Castles in the Middle

'Castles in the Middle' printed from https://nrich.maths.org/

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Castles in the Middle

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.

Getting started

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 this role.

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 team).

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 they worked.

Task Rules:

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 rules.
The team must take turns in asking questions.
The designer can only answer "Yes", "No" or give a number.

Designer guidelines:

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 it first.
You only answer questions of the agreed type.
You can only say "Yes" or "No", or give a number as an answer.

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 cube,
the colour of a cube in a particular position,
the symmetry properties of the castle,
any holes or gaps in the castle.

Observer Guidelines:

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 questions,
count the number of questions the team ask.

Why do this problem?

This task encourages the development of team-building skills such as sharing reasoning, allowing everyone to contribute and valuing those contributions, and coming to a consensus. 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. In addition learners are expected to use precision in describing the arrangement of the cubes.

Possible approach

If learners have never worked on this sort of problem before it is best to choose no more than six cubes, each of a different colour. You might also wish to add some additional constraints, such as:

all castles have at least one line of symmetry,
all castles have at least two turrets.

You will need a screen or cloth to hide the designer's arrangement. See **below.

Arrange learners in groups of four or five (five allows one to act as an observer).

You may wish to ask teams to record their questions before asking them. Then, as part of their review of the task, the team can discuss what may have been a more efficient set of questions to ask.

Cards for each role:

The rule cards for questioning can be found here: word, pdf.
The role card for designers can be found here: word, pdf.
The role card for observers can be found here: word, pdf.

When teams have finished working on the task it is important that they spend time discussing in groups, and then as a whole class, how well they worked as a team, what they have learned from the experience and what they would do differently next time. Your own observations, as well as those of observers might inform the discussions.

**You may prefer to prepare some images of an arrangement first to hand out to designers. Designers then answer the team questions, making a note of the number of questions asked. The team records each question before asking it. When the groups have finished you might discuss the number of questions used and ask the team with the fewest questions to share their ideas with the rest of the group.

Key questions

What things did you hear someone else say that you found really helpful?
How well did you listen to others in your group?
How easy was it to come to an agreement about the questions to ask?
Did everyone always agree and what did you do if you did not?

Possible extension

Increase the number of cubes available, so that there are multiple cubes of the same colour.

Learners may like to try one of the other 'All for one' tasks. Other skill-building tasks can be found by going to this article.

Possible support

Reduce the number of cubes available and make sure each is a different colour.

Allow the designer to give a colour as an answer.

Make a list of the allowed questions, such as:

How many cubes are there?
What colour is the cube to the left of the red cube?
Is the cube on top of/below the red cube blue?
Is there a cube behind/infront of the blue cube?