The 6-by-6 grid below can be partitioned into 6 L-shaped "hooks", nested inside one another:
- The largest hook is one full edge row plus one full edge column of the 6-by-6 grid (the two meet in a corner square), so it contains 11 squares.
- Remove it. What is left is a 5-by-5 square, and the next hook is one full edge row plus one full edge column of that square (9 squares).
- Continue in the same way. The hooks have 11, 9, 7, 5, 3 and 1 squares, and the smallest hook is a single square. Each hook may be oriented in any of its four possible ways, independently of the others.
Find where the hooks are. Then choose a different digit for each hook, using each of the digits 1, 2, 3, 4, 5, 6 exactly once, and place the digits so that the hook that gets digit k contains exactly k copies of k. The other squares of that hook are left empty. A hook's digit does not have to match its size (for example, the 9-square hook could hold five 5's), but a hook can only hold a digit k if it has at least k squares. In total, 1 + 2 + ... + 6 = 21 squares are filled and 15 are empty.
The filled squares must satisfy these rules:
- The filled squares form a single connected region. (Squares are connected if they are orthogonally adjacent.)
- Every 2-by-2 block of squares contains at least one empty square.
- Clues. In each row, read left to right, every maximal run of consecutive filled squares forms a number (for example, filled squares 3, 1, 2 in a row form the number 312, and a filled square with empty squares on both sides forms a one-digit number). The number outside the row is the greatest common divisor (GCD) of all the numbers in that row. Columns work the same way, reading top to bottom. Every row and every column contains at least one filled square.
c1 c2 c3 c4 c5 c6
5 3 1 2 2 44
+-----+-----+-----+-----+-----+-----+
r1 44 | | | | | | |
+-----+-----+-----+-----+-----+-----+
r2 1 | | | | | | |
+-----+-----+-----+-----+-----+-----+
r3 8 | | | | | | |
+-----+-----+-----+-----+-----+-----+
r4 2 | | | | | | |
+-----+-----+-----+-----+-----+-----+
r5 1 | | | | | | |
+-----+-----+-----+-----+-----+-----+
r6 1 | | | | | | |
+-----+-----+-----+-----+-----+-----+The clues as a list:
- Rows, top to bottom: r1 = 44, r2 = 1, r3 = 8, r4 = 2, r5 = 1, r6 = 1.
- Columns, left to right: c1 = 5, c2 = 3, c3 = 1, c4 = 2, c5 = 2, c6 = 44.
Question. In the completed grid, the empty squares fall into connected groups (again using orthogonal adjacency). What is the product of the areas (numbers of squares) of these groups?
Answer format. A single positive integer.