 123456789
A
B
C
D
E
F
G
H
I

### Sectors

Northwest No square with a given number in the Northwest Sector touches another square with a given number in the same sector along an edge. (In general we say a square touches another if it is the same square or it shares an edge or corner.) The sum of the given numbers in the North Sector is 13. There is a continuous path of touching squares with given numbers from the upper-right corner of the sudoku grid to the lower-left corner of the sudoku grid. The sum of the given numbers in the West Sector is 27, while their product is 1512. Rotational symmetry: if there is a given number in any square of the sudoku grid, there is also a (possibly different) given number in the square 180° opposite. The product of the given numbers in the East Sector is 288. Square I1 matches its antipodal square (A9) in the unsolved grid. The sum of the given numbers in the South Sector is 16. The given numbers in the Southeast Sector are exactly those in the East Sector.

### Rows

A The third square of Row A (A3) matches its antipodal square (I7) in the unsolved grid. The given numbers in Row B are exactly those in Column 6. The leftmost given number in Row C divides all the other given numbers in its sector. The product of the given numbers in D8 and D9 is 16. None of the squares with given numbers in Row E are directly above any of the four squares with given numbers in the row below, or directly below any of the squares with given numbers in the row above. The sum of the given numbers in Row F, which is the same as the sum of the given numbers in Row D, is 20. Row G, and every other row, has at least three given numbers. The product of the given numbers in H3 and H4 is 72. The sum of any given numbers in I1, I2, I3, and I4 is 6 more than the sum of any given numbers in H1 and H2, but 3 less than the sum of any given numbers in Column 5.

### Columns

1 The product of any given numbers in D1, E1, F1, and G1 is 63. The product of the given numbers in Column 2 is 18 more than the product of any given numbers in the Northeast Sector. The sum of the given numbers in Column 3 is 17. The three squares with given numbers in Column 4 are touching. There is only one given number in Column 5. There is only one square with a given number touching G6. None of the four squares with given numbers in Column 7 touch each other. There are the same number of squares with given numbers touching A8 as there are touching B8. The product of the given numbers in Column 9 is 180.

### Numbers

1 2 3 4 5 6 7 8 = A1 × B7 + C5 − ? = B2 × C5 + D2 − ? = C3 × D7 + E3 − ? = D4 × E9 + F3 − ? = E5 × F4 + G7 − ? = F6 × G1 + H5 − ? = G7 × H9 + I5 − ? = H8 × I3 + A3 − ? = I9 × A9 + B1 − ?