Inspired by my little cousin - the greatest artist of all time
One of the dreams of my little cousin is to become a great architect. His parents have given him many unit cube blocks to hone his skills.
One time, I noticed that he built 2 distinct square-based cuboids, in which exactly half of the cubes used could be seen from the outside.
What is the total number of cubes that he used?
Note: Cubes that are touching the ground, but not on the outside, canot be side. As an explicit example, the smallest building that hides at least 1 cube, has a square base of 3, and a height of 2. Exactly 1 cube not visible.
Special thanks to: Michael Mendrin
The answer is 3026.
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1 solution
Assuming Alisa's cousin aspires to be an architect, let us think of these cuboids as 'buildings' with f floors and analyse them floor by floor (horizontal cross sections).
Suppose the floor is N x N, enclosing a n x n 'core'. Clearly N = n + 2 giving a (4n + 4) 'boundary'.
Additionally the building also has a N x N 'roof'.
So the core of the building holds (n x n x f) cublets enclosed within a shell with (4n + 4)f + (n+2)^2 cublets.
Our goal is to make the number of cublets in the core and the shell equal. To compensate for the roof, each floor must have excess core cublets. This eliminates n = 1, 2, 3, 4 because n x n < 4n + 4
The next value n = 5 gives 25 core cublets per floor, while just 24 cublets in the boundary. So we gain 1 core cublet per floor. To compensate the 7 x 7 = 49 cublet roof, we need to have 49 floors, giving us a 7 × 7 × 5 0 = 2 4 5 0 cublet building.
Similarly, n = 6 gives 36 cublets core/floor and a 4 x 6 + 4 = 28 cublet boundary/floor. This gives 8 excess core cublets. To compensate the 8 x 8 = 64 cublet roof, we would need 64/8 = 8 floors. So the second building will have 8 × 8 × 9 = 5 7 6 cublets.
So the total number of cublets = 2450 + 576 = 3026