Tile and Trouble

Place square tiles in the 6 × 6 grid above so that the points in each row and each column add up to the number printed beside that row or below that column.

• A tile can have any side length: 1 × 1, 2 × 2, 3 × 3, and so on. Each tile must lie along the grid lines, entirely inside the grid.
• Every cell of an n × n tile is worth n points. Cells not covered by a tile are worth 0.
• Tiles may not overlap. Not every cell has to be covered.

The clues in text form (rows R1–R6 from top to bottom, columns C1–C6 from left to right):

There is exactly one way to place the tiles. Once the grid is solved, the uncovered cells fall into empty regions: two empty cells are in the same region if you can walk from one to the other through empty cells that share an edge. A single isolated empty cell is a region of area 1.

Answer format: a single integer, the product of the areas of all the empty regions.