Cubic Vision
How many cubes can you see?
Age
14 to 16
Challenge level
Problem
A wooden cube measuring $11$cm by $11$cm by $11$cm is made from $1$cm by $1$cm by $1$cm cubes.
What is the greatest number of centimetre cubes visible from any point?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
One can see the greatest number of cubes when looking at three faces at once.
We count the cubes on each face, giving $3\times 11^2=363$ cubes, but have to subtract from this the cubes along the three edges that have been counted twice, and then add back the cube at the corner (which has three faces visible).
The final quantity is $363-(3\times 11)+1=331$ cubes.