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Find the length along the shortest path passing through certain points on the cube.

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Shaping the Universe I - Planet Earth

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The Perforated Cube

Age 14 to 16 Challenge Level:
A large cube made from 125 smaller cubes, 5 by 5 by 5, is 'perforated' by removing cubes.

A 125 cube sparsely made of small floating cubes

Any cube may be removed so imagine some uncoloured, transparent cubes propping up anything that needs it.

If a 'perforated cube' has the three views (projections) below, what is the most and the least cubes possible to make a shape that has these three projections ? In other words what could be the most and the least cubes left from the original 125 ?