Optimal filling of grid figure with squares

Question

recently I have designed a puzzle for children to solve. However I would like to now the optimal solution.

The problem is as follows: You have this figure made up of

Answer 1

(This is not meant to be a full answer; I'm just sharing what I'm working on so that we can collaborate.)

I think a good first step is to transform the binary grid by giving every cell the value of the maximum size of square that the cell can be the top-left corner of, like this:

0,0,3,2,1,0,3,2,2,2,2,1,0,0,0,0,0,0,2,1,0,0,0,0,0,2,1,0,0,0
0,0,2,2,2,3,3,2,1,1,1,1,0,0,0,3,3,3,3,3,3,2,1,0,0,1,2,1,0,0
0,2,1,1,1,2,3,2,1,0,0,0,0,3,2,2,2,2,2,2,3,3,2,1,0,0,3,2,1,0
3,2,1,0,0,1,3,2,1,0,0,0,3,2,2,1,1,1,1,1,2,3,3,2,1,0,2,2,2,1
3,3,2,1,0,0,2,2,2,1,0,3,2,2,1,1,0,0,0,0,1,2,4,3,2,2,1,1,1,1
2,3,3,2,1,0,2,1,1,1,2,3,2,1,1,0,0,0,0,0,0,1,3,3,2,1,1,0,0,0
1,2,3,4,3,2,1,1,0,0,1,3,2,1,0,0,0,0,0,0,0,0,2,2,2,1,0,0,0,0
0,1,2,3,3,2,1,0,0,0,0,2,2,1,0,0,0,0,0,0,0,0,2,1,1,2,2,2,1,0
0,0,1,2,3,2,1,0,0,0,0,1,2,1,0,0,0,0,0,0,0,0,2,1,0,1,1,2,1,0
0,0,0,1,2,2,1,0,0,0,0,0,2,1,0,0,0,0,0,0,0,2,1,1,0,0,0,3,2,1
1,0,0,0,1,2,1,0,0,0,0,0,4,3,2,1,0,0,0,4,3,2,1,0,0,0,0,2,2,1
2,1,0,0,0,1,2,1,0,0,5,5,4,4,4,4,4,4,4,5,5,4,3,2,1,0,0,1,2,1
3,2,1,0,0,0,1,6,6,5,4,4,4,3,3,3,3,3,3,4,4,5,4,3,2,1,0,0,1,1
3,2,1,0,0,0,0,6,5,5,4,3,3,3,2,2,2,2,2,3,3,4,5,4,3,2,1,0,0,0
3,2,2,2,2,7,6,6,5,4,4,3,2,2,2,1,1,1,1,2,2,3,5,5,4,3,2,1,0,0
2,2,1,1,1,7,6,5,5,4,3,3,2,1,1,1,0,0,0,1,1,2,4,6,5,4,3,2,1,0
2,1,1,0,0,7,6,5,4,4,3,2,2,1,0,0,0,0,0,0,0,1,3,6,5,4,3,2,1,0
1,1,0,0,8,7,6,5,4,3,3,2,1,1,0,0,0,0,0,0,0,0,2,7,6,5,4,3,2,1
1,0,0,0,8,7,6,5,4,3,2,2,1,0,0,0,0,0,0,0,0,0,1,7,6,5,4,3,2,1
0,0,0,7,8,7,6,5,4,3,2,1,1,0,0,0,0,0,0,0,0,0,0,6,6,5,4,3,2,1
0,0,0,6,8,7,6,5,4,3,2,1,0,0,0,0,0,0,0,0,0,0,0,6,5,5,4,3,2,1
0,0,0,5,7,7,6,5,4,3,2,1,0,0,0,0,0,0,0,0,0,0,0,6,5,4,4,3,2,1
0,0,0,4,6,7,7,6,5,4,3,2,1,0,0,0,0,0,0,0,0,0,6,5,5,4,3,3,2,1
0,0,0,3,5,6,7,7,6,5,4,3,2,1,0,0,0,0,0,0,0,6,6,5,4,4,3,2,2,1
1,0,0,2,4,5,6,7,8,7,6,5,4,3,2,1,0,0,0,7,6,6,5,5,4,3,3,2,1,1
1,0,0,1,3,4,5,6,7,7,8,8,8,8,8,8,7,7,6,6,6,5,5,4,4,3,2,2,1,0
2,1,0,0,2,3,4,5,6,6,7,7,8,7,7,7,7,6,6,5,5,5,4,4,3,3,2,1,1,0
2,1,0,0,1,2,3,4,5,5,6,6,8,7,6,6,6,6,5,5,4,4,4,3,3,2,2,1,0,0
3,2,1,0,0,1,2,3,4,4,5,5,8,7,6,5,5,5,5,4,4,3,3,3,2,2,1,1,0,0
3,2,1,0,0,0,1,2,3,3,4,4,8,7,6,5,4,4,4,4,3,3,2,2,2,1,1,0,0,0
4,3,2,1,0,0,0,1,2,2,3,3,8,7,6,5,4,3,3,3,3,2,2,1,1,1,0,0,0,0
3,3,2,1,0,0,0,0,1,1,2,2,8,7,6,5,4,3,2,2,2,2,1,1,0,0,0,0,0,0
2,2,2,2,1,0,0,0,0,0,1,1,8,7,6,5,4,3,2,1,1,1,1,0,0,0,0,0,0,0
1,1,1,2,1,0,0,0,0,0,0,0,8,7,6,5,4,3,2,1,0,0,0,0,0,0,0,0,0,0
0,0,0,2,1,0,0,0,0,0,0,0,8,8,7,6,5,4,3,2,1,0,0,0,0,0,0,0,0,0
0,0,0,2,1,0,0,0,0,0,0,6,8,7,7,6,6,5,4,3,2,1,0,0,0,0,0,0,0,0
0,0,0,2,2,2,3,3,3,3,3,5,7,7,6,6,5,5,4,3,3,3,3,2,1,0,0,0,0,0
0,0,3,2,1,1,3,2,2,2,2,4,6,6,6,5,5,4,4,3,2,2,2,2,1,0,0,0,0,0
0,0,3,2,1,0,3,2,1,1,1,3,5,5,5,5,4,4,3,3,2,1,1,2,1,0,0,0,0,0
0,0,3,2,1,0,3,2,1,0,0,2,4,4,4,4,4,3,3,2,2,1,0,2,1,0,0,0,0,0
0,4,3,2,1,0,3,2,1,0,0,1,3,3,3,4,3,3,2,2,1,1,0,2,1,0,0,0,0,0
0,4,3,2,1,0,3,2,1,0,0,0,2,2,2,3,3,2,2,1,1,0,0,2,1,0,0,0,0,0
0,4,3,2,1,0,3,2,1,0,0,0,1,1,1,2,2,2,1,1,0,0,0,2,1,0,0,0,0,0
3,3,3,2,1,0,3,2,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0,3,2,1,0,0,0,0
2,2,2,2,1,0,2,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,1,0,0,0,0
1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0

If you wanted to go through every option using brute force, you'd try every size of square that a cell could be the corner of (including 1x1), mark the square with zeros, adjust the values of the cells up to 7 places left/above the square, and recurse with the new grid.

If you iterated over the cells top-to-bottom and left-to-right, you'd only have to copy the grid starting from the current row to the bottom row, and you'd only have to adjust the values of cells up to 7 places to the left of the square.

The JS code I tested this with is fast for the top 2 or 3 rows of the grid (result: 24 and 44), takes 8 seconds to finish the top 4 rows (result: 70), and 30 minutes for 5 rows (result: 86). I'm not trying 6 rows.

But, as you can see from this grid, the number of possibilities is so huge that brute force will never be an option. On the other hand, trying something like adding large squares first, and then filling up the leftover space with smaller squares, is never going to guarantee the optimal result, I fear. It's too easy to come up with examples that would thwart such a strategy.

7,6,5,4,3,2,1,0,0,0,0,0,0,7,6,5,4,3,2,1
6,6,5,4,3,2,1,0,0,0,0,0,0,6,6,5,4,3,2,1
5,5,5,4,3,2,1,0,0,0,0,0,0,5,5,5,4,3,2,1
4,4,4,4,3,2,1,0,0,0,0,0,0,4,4,4,4,3,2,1
3,3,3,3,3,2,1,0,0,0,0,0,0,3,3,3,3,3,2,1
2,2,2,2,2,2,1,0,0,0,0,0,0,2,2,2,2,2,2,1
1,1,1,1,1,1,8,7,6,5,4,3,2,1,1,1,1,1,1,1
0,0,0,0,0,0,7,7,6,5,4,3,2,1,0,0,0,0,0,0
0,0,0,0,0,0,6,6,6,5,4,3,2,1,0,0,0,0,0,0
0,0,0,0,0,0,5,5,5,5,4,3,2,1,0,0,0,0,0,0
0,0,0,0,0,0,4,4,4,4,4,3,2,1,0,0,0,0,0,0
0,0,0,0,0,0,3,3,3,3,3,3,2,1,0,0,0,0,0,0
0,0,0,0,0,0,2,2,2,2,2,2,2,1,0,0,0,0,0,0
7,6,5,4,3,2,1,1,1,1,1,1,1,7,6,5,4,3,2,1
6,6,5,4,3,2,1,0,0,0,0,0,0,6,6,5,4,3,2,1
5,5,5,4,3,2,1,0,0,0,0,0,0,5,5,5,4,3,2,1
4,4,4,4,3,2,1,0,0,0,0,0,0,4,4,4,4,3,2,1
3,3,3,3,3,2,1,0,0,0,0,0,0,3,3,3,3,3,2,1
2,2,2,2,2,2,1,0,0,0,0,0,0,2,2,2,2,2,2,1
1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,1

In the above example, putting an 8x8 square in the center and four 6x6 squares in the corners gives a lower score than putting a 6x6 square in the center and four 7x7 squares in the corners; so a greedy approach based on using the largest square possible will not give the optimal result.

---

This is how far I got by isolating zones connected by corridors of maximum width 3, and running the brute-force algorithm on the smaller grids. Where the border has no orange zone, adding another 2 cells doesn't increase the score of the isolated zone, so those cells can be used by the main zone unconditionally.