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Look Inside (1)


Look Inside

Imagine you have an unlimited number of small unit cubes (all the same size 1×1×1). From these unit cubes, you start to build bigger and bigger cubes in such a way that a cube will be wrapped into other unit cubes. This “unit cube wrap” can be called a layer. Then you build cube C of the size of 5×5×5 unit cubes and try to answer the following questions:

a)    How many layers of cube C do you have to unwrap to get to one single unit cube?

b)    How many unit cubes does each layer have? (Start with the visible (outer) layer and write the answer in the form "'number' and 'number' and 'number' and etc.")

c)     How many unit cubes are hidden in cube C that cannot be seen at all?

d)    How many unit cubes of the visible layer touch the faces of unit cubes of the previous layer?

e)     Remove the unit cubes from cube C that have just three touching faces with the other unit cubes. How many unit cubes remain in the visible layer?

You can use the following GeoGebra Applet with the printed "Navigation Guide" to solve the problem. Applet Instructions: a) Use the sliders to look inside cube C (1st PRISM - 5th PRISM) b) Use 'FRONT VIEW OF A CUBE' to hide/show cubes in 1st PRISM - 3rd PRISM. c) Click on the individual cubes to change their colour (one click - yellow colour, two clicks - red colour, three clicks - green colour). FEEL FREE TO BE CREATIVE IN YOUR SOLUTION :-)

Navigation Guide