The Red 9: The most constrained number on the board is the Red 9. As the only possible perfect square of that size, it must be a 3x3 shape. Its position in the bottom-right quadrant gives it little wiggle room. Placing it flush into the bottom-right corner is the only logical move, as any other placement would create unusable 1-cell or 2-cell gaps against the edges. This single move establishes a huge, immovable block.

The Purple 4: With the 3x3 Red 9 in place, the Purple 4 above it is now constrained. It could be a 1x4 strip or a 2x2 square. However, a 1x4 vertical strip is impossible, as it would collide with the Red 9. A 1x4 horizontal strip would run into the Teal 2. Therefore, the Purple 4 must be a 2x2 square, which fits neatly above the Red 9.

The Teal 2: The placement of the Red 9 and Purple 4 creates a narrow vertical channel one cell wide. The Teal 2, which must be a 1x2 or 2x1 shape, is now forced into its only possible configuration: a 1x2 vertical strip.

The Blue 6: Now look at the top-right corner. The Blue 6 could be 1x6, 6x1, 2x3, or 3x2. But our previous moves have already defined the grid's constraints. A 6x1 or 3x2 is impossible due to the height established by the Red 9 (3 cells) and Purple 4 (2 cells). A 2x3 shape would collide with the newly placed Teal 2. This leaves only one option: the Blue 6 must be a 1x6 horizontal strip spanning the entire top of the puzzle.

This top strip now creates a "ceiling" for the Orange 3, forcing it into a 1x3 horizontal shape. This, in turn, helps define the space for the Grey 4 in the bottom-left corner. There is no longer room for a 2x2 square, so it must be a 1x4 vertical strip. This final move creates perfect 2x2 pockets for the Yellow 4 and the Green 4, completing the grid.