The most starting move had nothing to do with a single shape. First, we identify the grid size: 8x8, for a total area of 64 cells. Next, we sum the known numbered clues: 4+12+4+6+6+4+12+3 = 51. The total area for the two unnumbered squares (Light Blue and Dark Blue) must be the difference: 64 - 51 = 13 cells. The only two perfect squares that sum to 13 are 4 (a 2x2 square) and 9 (a 3x3 square). With this single calculation, we've defined the exact size of our two most ambiguous pieces before placing a single one. This is a important practice for complex grids.

A subtle but critical clue was the Orange 4. Since 4 is a perfect square, this piece was constrained to be a 2x2 square, not a 1x4 strip. We now had three squares to place: a 3x3, a 2x2, and another 2x2.

The Pink 3 in the bottom-left corner was the perfect physical starting point. As a prime number, its dimensions can only be 1x3 or 3x1. Placed in a corner, it cannot be a 1x3 vertical strip (as that would leave an unfillable 2-cell gap beneath it). Therefore, it must be a 3x1 strip along the bottom edge.

Placing the Pink 3x1 immediately forces the Purple 4 above it. Confined to the left edge and boxed in by the Pink 3, its only possible fit is a 1x4 vertical strip.

This leaves a 3-cell wide space in the top-left. Our 3x3 square (the Light Blue one) is the only piece that fits this space. Placing it defines the entire top-left quadrant.

The placement of the Light Blue 3x3 and the Red 4 (which must be a 4x1 strip in its corner) creates a 4x3 empty space between them, which is a containment for the Green 12.