I told you that the best solution leaves only 3 boxes empty. This means that at least one row and one column will have all four shapes. Pick a row as your starting position; I chose the bottom row. The sequence of the objects in the starter row doesn't matter, so I kept the four objects as they were in the puzzle statement.

One column will have all four shapes. I chose to try one of the center columns because I chose an edge row to have four objects. This guess might not work out, but it seems likely to work better than an edge. I'll try to populate the third column.

The shape above the diamond must be a triangle. All the other shapes share a diagonal or column with a shape on the bottom row. The remaining two shapes in that column are a square and a circle, but we don't know the right order yet.

There is only one remaining box that could contain a triangle (the second box on the top row). Notice that the triangles are placed a knight's move away from each other. A knight's move is a chess maneuver that steps two squares in one direction and then turns 90 degrees to step one square to the side.