I like these sorts of things every so often, so thought I'd throw this up if anyone wanted to have a stab at it
Let's consider "stacks" of blocks in any number of dimensions - and for this example we'll consider a side length of 5.
So, a one dimensional stack of blocks would just be 5 blocks on top of each other - therefore using 5 blocks in total.
A two dimensional "stack" of blocks would have 5 blocks on the bottom, then 4, then 3, then 2 and then 1. So this would use 15 blocks in total.
A 3 dimensional "stack" of blocks would be a triangular pyramid. So it'd have 15 blocks on the bottom arranged in a triangle (5, 4, 3, 2, 1) then 10 blocks arranged in a triangle (4, 3, 2, 1) all the way up to just one block at the top. So this would use 35 blocks in total.
Applying this same logic, how many blocks would be needed to make a 4 dimensional "stack" of side length 5?
Following on from this, can you write a method / function that will, given the side length and number of dimensions, work out the total number of blocks needed?
An interesting spin off of this problem is this - given a normal triangular pyramid of blocks (3D) and assuming each block can only take a certain amount of weight before it collapses, if built tall enough will the whole pyramid collapse under its own weight? If we apply this to a 4D "pyramid", would it do the same thing? If so, is there a number of dimensions for which it won't ever collapse under its own weight?
EDIT: For clarification, I'm still assuming gravity is a vector force only ever acting on a single dimension.