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Peeling the Apple or the Cone That Lost Its Head
Age 14 to 16
Challenge Level 
This is a nice activity to take slowly and build up understanding in each part.
It also makes a nice lead in towards maths at the next level, especially if the spreadsheet is used to improve accuracy as each fustum height becomes smaller without limit.
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F'arc'tion
At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.
