### 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?

### Three Squares

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

### 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?

##### Stage: 1 Challenge Level:

Excellent solutions to this problem were sent in by William from All Saints, Carshalton and Ruth from Swanbourne House School.

William says that the black circle could be a sphere, hemisphere, cylinder or cone.

Ruth explains that the square could be a cube, a cuboid on its end or a square-based pyramid.

The triangle could be a triangular prism, tetrahedron, one side of any shaped pyramid or a cone according to William.

Finally, Ruth says that the rectangle could be any kind of prism, a cylinder or a cuboid. William also suggests it could be a rectangular based pyramid.

Stephen who is home educated also said that the square could be a triangular prism (with the non-triangle bits being squares) and in fact any prism with square faces.

Thomas and Nathan from Our Lady and St Werburg; Ebony, Lucy and Carys from Eaton Park Primary and Armitage Class from East Hoathly also sent in many of the above suggestions.

Jonathan from King David High School wrote in with some suggestions of items which could make the shadows:

The circular shape could be an enlarged smarties tube from the side.
The square shape could be the side view of a die.
The triangular shape could be a view of a tolberone box.
The rectangular shape could be a view of a biscuit box.

This is another way to look at the problem - thank you Jonathan.