The Perforated Cube
A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?
Problem
A large cube made from 125 smaller cubes, 5 by 5 by 5, is 'perforated' by removing 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 ?
Getting Started
Student Solutions
Edison from Shatin School included some edited versions of the diagram given in the hints to support his argument:
Then you can take away blocks, checking each face projection so its unchanged.
On the far E, you can take away 4 on the top prong, 4 on the bottom prong, and the 1 back block on the middle prong. The middle of the S cannot be removed as it is needed for the S face. The on the close E you can take 4 from the middle prong, and then the back block on the top and bottom prong.
So we have removed $15$ blocks, and you cannot remove any more. So the minimum total is $41-15=26$
Well done Edison, can anyone think of any other interesting projections to aim for?
Teachers' Resources
This problem provides a rich context for visualisation and suggests many similar lines of enquiry.
For example :
- Can all letters of the alphabet be represented using only a 5 by 5 array of cubes ?
- Is it possible to create every combination of three letters in a 'perforated cube' ?
- What difference does the orientation of the letters make ?
- How many ways are there to orientate three non-symmetric letters such as F, J and P ?
- What difference will it make to use letters that have symmetry of some kind ?
- Can every solution arrangement have either more cubes added or some removed, and still be a solution arrangement ?
- Is there a relationship between the maximum and minimum number of cubes for any solution ?
Exploring the perforated cube problem, and related questions, using clip-together plastic cubes will provide invaluable 'concrete' sensory experience for students as they stretch their powers of visualisation, express the problem, test their conjectures, or represent their solutions.
The information above is repeated on the Hint page which also contain this excellent video of one solution for E, S and H - video of an ESH solution