Question

I’m working on a spatial stacking problem… at the moment I’m trying to solve in 2D but will eventually have to make this work in 3D.

I divide up space into n x n squares around a central block, therefore n is always odd… and I’m trying to find the number of locations that a rectangle of any dimension less than n x n (eg 1×1, 1×2, 2×2 etc) can be placed, where the middle square is not available.

So far I’ve got this..

total number of rectangles = ((n^2 + n)^2 ) / 4

..also the total number of squares = (n (n+1) (2n+1)) / 6

However I’m stuck in working out a formula to find how many of those locations are impossible as the middle square would be occupied.

So for example:

[] [] []

[] [x] []

[] [] []

3 x 3 board… with 8 possible locations for storing stuff as mid square is in use.
I can use 1×1 shapes, 1×2 shapes, 2×1, 3×1, etc…

Formula gives me the number of rectangles as: (9+3)^2 / 4 = 144/4 = 36 stacking locations
However as the middle square is unoccupiable these can not all be realized.

By hand I can see that these are impossible options:

1×1 shapes = 1 impossible (mid square)
2×1 shapes = 4 impossible (anything which uses mid square)
3×1 = 2 impossible
2×2 = 4 impossible
etc
Total impossible combinations = 16

Therefore the solution I’m after is 36-16 = 20 possible rectangular stacking locations on a 3×3 board.

I’ve coded this in C# to solve it through trial and error, but I’m really after a formula as I want to solve for massive values of n, and also to eventually make this 3D.

Can anyone point me to any formulas for these kind of spatial / tessellation problem?
Also any idea on how to take the total rectangle formula into 3D very welcome!

Thanks!

Answer 1

Ok.. so I’ve got an answer now which is.. the total impossible cases is defined by:

n^4 where n is the order of grid size using only odd grids

2^4 = 16 (grid is 3 by 3)
3^4= 81 (grid is 5 by 5)
4^4 = 256 (grid is 7 by 7)
etc