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Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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Three Squares

What is the greatest number of squares you can make by overlapping three squares?

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A City of Towers

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Start Cube Drilling

Stage: 1 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The whole idea of this problem is to invite children to picture something in their mind and in this instance, pupils will need to be familiar with properties of a cube. Ideally, it would be good to encourage your class to tackle this challenge purely by trying to imagine what is happening. To convince you and each other of their solutions, they will need to explain particularly carefully what they are picturing, which can be quite tricky, and you may find that they gesticulate rather a lot! In order to reach a joint conclusion, you might find it helpful to make a model of the cube from interlocking cubes.

To introduce the activity, it might be worth simplifying the problem by starting with a smaller cube (say 2 by 2 by 2). Invite them to picture this cube in their heads and ask questions, such as:

  • How many small cubes are there altogether? [8]
  • How many corner cubes? [8]
  • What other kinds of small cubes make up the cube? [none - they are all corner cubes]
You could then move on to picturing a 3 by 3 by 3 cube and probe in a similar way before trying this problem.

If your class would like more of a challenge having tried this problem, they could look at Cube Drilling .

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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NRICH is part of the family of activities in the Millennium Mathematics Project.