Fill every cell of the 6-by-6 grid below, including the cells that show grey numbers, with a digit from 1 to 6 so that each row, each column, and each outlined 2-by-3 region (two rows tall, three columns wide) contains each of the digits 1 to 6 exactly once.
Each grey number is a clue. It equals the product of two of the cells orthogonally adjacent to the grey cell (directly above, below, left or right of it) in the completed grid. The digit written in a grey cell is unrelated to its own clue, but it may itself be one of the factors of a neighbouring clue.
In the diagram, (n) marks a grey cell with clue n, and . marks an ordinary empty cell. The six regions are separated by the | line between columns 3 and 4 and by the = lines below rows 2 and 4.
C1 C2 C3 C4 C5 C6
+------------------+------------------+
R1 | . . (9) | . . . |
R2 | . (16) (36) | . (36) . |
+==================+==================+
R3 | . . (25) | . . . |
R4 | . . . | (16) (4) . |
+==================+==================+
R5 | . (1) . | . . . |
R6 | . . . | . . . |
+------------------+------------------+The eight clues, as a list (row, column): R1C3 = 9, R2C2 = 16, R2C3 = 36, R2C5 = 36, R3C3 = 25, R4C4 = 16, R4C5 = 4, R5C2 = 1.
Question. What is the sum of the squares of the digits written in the eight grey cells of the completed grid?
Answer format. A single positive integer.
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