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### Number and algebra

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# Octa-flower

Start with a thought experiment in which you take some regular octahedra and, on each one, colour two faces that meet at a vertex but not along an edge. Now imagine taking two octahedra and gluing them together with coloured faces in contact so that the vertices where the coloured faces meet coincide. Now glue another octahedron on, coloured face to coloured face, so that the three vertices where the coloured faces meet coincide. Continue in the same way as long as possible. Calculate how many octahedra can be joined together in this way.

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Age 16 to 18

Challenge Level

Start with a thought experiment in which you take some regular octahedra and, on each one, colour two faces that meet at a vertex but not along an edge. Now imagine taking two octahedra and gluing them together with coloured faces in contact so that the vertices where the coloured faces meet coincide. Now glue another octahedron on, coloured face to coloured face, so that the three vertices where the coloured faces meet coincide. Continue in the same way as long as possible. Calculate how many octahedra can be joined together in this way.

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.