bad dailyprog solution

My solution is somewhat trivial, but with some thought and intuition behind it. Just a theory, not a very good one - no mathematical proofs.

I started with a 2x2 matrix. There are 3 unique optimized solutions for a 2x2. When I say unique, I mean that there is no way to, through any combination of rotation, mirroring, or transposition, to turn one solution into another.

They are:

23

14

------------

24

13

-----------

32

14

Now, expanding into a 3x3, I thought to start with one of the 3 2x2 solutions. In order to add a 5, it seemed to me like I would have to add so much scaffolding in order to fulfill the requirements that it would negate the potential gains of a 5, and be less efficient. /u/Godspiral has a couple of 3x3s which contain a 5, which are nearly maximized but ultimately fall short of the (theorized?) single unique solution:

424

313

424

 

From n=2 to n=3 I saw one of the chiral solutions to n=2 patterned to create the solution to n=3. I theorize that this isn't a coincidence; that the n=2 solution, being the smallest chunk of a complete optimized square, when arranged properly and patterned, may result in the highest sum for any m by n grid. this follows the pattern

4242424.....(2 if n is even, 4 if odd)

3131313....(1 if n is even, 3 if odd)

............

(3,1 row if m is even, 4,2 row if odd)

As such my current solution is:

    def grid(n, m=None):
        m = n if m is None else m
        pattern = [[4,2], [3,1]]
        board = []
        board_sum = 0
        for y in range(n):
            row = []
            for x in range(m):
                row.append(pattern[y%2][x%2])
            board.append(row)
            board_sum += sum(row)
        return board_sum, board

For which the square n x n sums are as follows:

n | sum
-------|---------
2 | 10
3 | 27
4 | 40
5 | 70
6 | 90
7 | 133
8 | 160
9 | 216

I gladly await being proven wrong =)