Reply To: How many squared Columns can be arranged on a squared Parking Level using the “Array Modifier” without any Intersections?
@duerer nice question!
For the answer, lets first look at a related, but simpler problem:
we will try and divide a side of 36 (72 divided by the length of the default cube) in equal parts. To do this, we look at the prime factorization of 72:
2x2x3x3. Consider all possible combinations of those primes and their product (including the ’empty’ product, where none of them is being multiplied, resulting in 1).
We get: 1, 2, 3, 4, 6, 9, 12, 18 and 36. These are actually pairs of whole number solutions to the equation : A x B = 36.
So we have 9 possible divisions, for instance:
But the two corners aren’t occupied. If we add one more cube:
that will lie exactly outside of our garage.
If our Plane were one cube larger, our problem would be solved. And this already gives us the solution to the original problem:
We divide 72 -2 = 70 (side length minus one cube ) by 2 (cube length) and get 35.
Then look at the prime factorization: 35 = 5 x 7. This yields two possibilities:
Array count of 6 (5 plus 1!), with a Relative Offset of 7 and an Array count of 8 (7 plus 1) with a Relative Offset of 5.
