Make a cube out of straws and have a go at this practical
Reasoning about the number of matches needed to build squares that
share their sides.
How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!
Thomas and Daniel have sent this almost
complete solution. Well done.
I have omitted a part of the solution where they talked about
the icosahedron being able to fit into three triangular hollows. I
think they were trying to say that three of the five platonic
solids have triangular faces so the toy would have three triangular
hollows. This would mean that the triangle could fit the toy $3
\times 12$ ways, the octahedron $3 \times 24$
ways and the isocahedron $3 \times 60$ ways - that
complicates things a little, but then of course there are the
number of ways you can arrange the three solids with triangular
faces on the toy (would that matter?). So the number of times the
bell rings wasn't quite correct but I think this part of the
question is open to some interpretation anyway.
The tetrahedron can fit into the triangular hollow twelve
different ways as it has four sides and for each side three
different ways of fitting in.
The cube having six sides in the shape of squares can fit into
the square hollow in four different ways and so fits twenty-four
The octahedron has eight sides in triangles and so fits in
twenty-four different ways.
The dodecahedron with twelve sides in pentatonic shapes fits
The icosahedron can fit into the triangular hollow sixty
Add all the ways that each fits into the suited hollow and you
find the number of times the bell rings.
Andy emailed us and suggested that there could
be $24$ ways of fitting the tetrahdron into the hollows if you
could put the tetrahedron in "vertex first". This might be possible
if the hollow was deep enough. Thank you for drawing this to our