Background
Recently I was doing some recreational mathematics and stumbled across an interesting observation:
What is the maximum number of diamonds (square rotated at $45^o$) can one fit into an $n \times n$ square, where the area of the diamond is $2$? The picture below illustrates a square of $n=3$ and fits $1$ diamond
It shouldn't be hard to convince yourself that this is related to the question of how many numbers are there divisible by $2$ less than $n^2$. Why? Consider labelling each grid point with a number:
A diamond is formed whenever one joins the factors of $2$ together! Of course,this is only true when $n$ is an odd number $n>1$.
(Also imagine you have the number of diamonds you can fit then with Euler's characteristic formula you can find the number of vertices)
Now, we can extend this for a diamonds in a $5 \times 5$ square:
Similarly for a cube we can imagine divide it into smaller cubes and labelling them: $1,2,3,4,\dots$ and join the mid-point of every $2$'nd brick and get a pyramid. which happens to be maximum packing efficiency of the pyramids in a cube.
Conjecture
"The packing efficiency always is maximum for a shape created by joining the mid-point vertices in a $n$-dimensional diamond (square for $2$, pyramid for $3$, $4 =$ ?)"
Question
Is there a general formula for the shape (note in the pyramid the tip is the centre of the cube rather than the midpoint of the edge)? Is the conjecture true or false?