With this solving technique, we will prove the two following assertions:
Each assertion is proved by a chain of simple assertions. The chains may be dynamic, which means that the conclusions of multiple sub-chains are combined.
The details of each chain are given below. Use the view selector below the grid to switch between the graphical illustrations of the two chains.
Chain 1: If A1 contains the value 9, then I9 must contain the value 4 (View 1):
(1) If A1 contains the value 9 then A1 cannot contain the value 1 (the cell can contain only one value)
(2) If A1 does not contain the value 1 then A3 must contain the value 1 (only remaining position in the box)
(3) If A3 contains the value 1 then A3 cannot contain the value 6 (the cell can contain only one value)
(4) If A3 does not contain the value 6 then C3 must contain the value 6 (only remaining position in the box)
(5) If C3 contains the value 6 then C9 cannot contain the value 6 (the value can occur only once in the col)
(6) If C9 does not contain the value 6 then B9 must contain the value 6 (only remaining position in the box)
(7) If B9 contains the value 6 then B4 cannot contain the value 6 (the value can occur only once in the col)
(8) If B4 does not contain the value 6 then B4 must contain the value 4 (only remaining potential value in the cell)
(9) If B4 contains the value 4 then D4 cannot contain the value 4 (the value can occur only once in the row)
(10) If A1 does not contain the value 1 (misplaced towel) then E1 must contain the value 1 (only remaining position in the row)
(11) If E1 contains the value 1 then F2 cannot contain the value 1 (the value can occur only once in the box)
(12) If A3 contains the value 1 (2) then F3 cannot contain the value 1 (the value can occur only once in the row)
(13) If F3 does not contain the value 1 and F2 does not contain the value 1 (11) then F6 must contain the value 1 (only remaining position in the column)
(14) If F6 contains the value 1 then D4 cannot contain the value 1 (the value can occur only once in the box)
(15) If D4 does not contain the value 1 and D4 does not contain the value 4 (9) then D4 must contain the value 8 (only remaining potential value in the cell)
(16) If D4 contains the value 8 then E5 cannot contain the value 8 (the value can occur only once in the box)
(17) If D4 contains the value 8 (15) then D5 cannot contain the value 8 (the value can occur only once in the box)
(18) If F6 contains the value 1 (13) then F6 cannot contain the value 7 (the cell can contain only one value)
(19) If F6 does not contain the value 7 then F5 must contain the value 7 (only remaining position in the box)
(20) If F5 contains the value 7 then F5 cannot contain the value 8 (the cell can contain only one value)
(21) If B9 contains the value 6 (6) then B9 cannot contain the value 2 (the cell can contain only one value)
(22) If B9 does not contain the value 2 then G9 must contain the value 2 (only remaining position in the row)
(23) If G9 contains the value 2 then H7 cannot contain the value 2 (the value can occur only once in the box)
(24) If F6 does not contain the value 7 (18) then H6 must contain the value 7 (only remaining position in the row)
(25) If H6 contains the value 7 then H6 cannot contain the value 2 (the cell can contain only one value)
(26) If H6 does not contain the value 2 and H7 does not contain the value 2 (23) then H5 must contain the value 2 (only remaining position in the column)
(27) If H5 contains the value 2 then H5 cannot contain the value 8 (the cell can contain only one value)
(28) If H5 does not contain the value 8, F5 does not contain the value 8 (20), D5 does not contain the value 8 (17) and E5 does not contain the value 8 (16) then I5 must contain the value 8 (only remaining position in the row)
(29) If I5 contains the value 8 then I9 cannot contain the value 8 (the value can occur only once in the col)
(30) If F5 contains the value 7 (19) then F5 cannot contain the value 5 (the cell can contain only one value)
(31) If F6 contains the value 1 (13) then F6 cannot contain the value 5 (the cell can contain only one value)
(32) If F6 does not contain the value 5 and F5 does not contain the value 5 (30) then F8 must contain the value 5 (only remaining position in the column)
(33) If F8 contains the value 5 then F8 cannot contain the value 9 (the cell can contain only one value)
(34) If F8 does not contain the value 9 then F9 must contain the value 9 (only remaining position in the box)
(35) If F9 contains the value 9 then I9 cannot contain the value 9 (the value can occur only once in the row)
(36) If I9 does not contain the value 9 and I9 does not contain the value 8 (29) then I9 must contain the value 4 (only remaining potential value in the cell)
Chain 2: If A1 contains the value 9, then I9 cannot contain the value 4 (View 2):
(1) If A1 contains the value 9 then A1 cannot contain the value 1 (the cell can contain only one value)
(2) If A1 does not contain the value 1 then E1 must contain the value 1 (only remaining position in the row)
(3) If E1 contains the value 1 then F2 cannot contain the value 1 (the value can occur only once in the box)
(4) If A1 does not contain the value 1 (misplaced towel) then A3 must contain the value 1 (only remaining position in the box)
(5) If A3 contains the value 1 then F3 cannot contain the value 1 (the value can occur only once in the row)
(6) If F3 does not contain the value 1 and F2 does not contain the value 1 (3) then F6 must contain the value 1 (only remaining position in the column)
(7) If F6 contains the value 1 then F6 cannot contain the value 7 (the cell can contain only one value)
(8) If F6 does not contain the value 7 then F5 must contain the value 7 (only remaining position in the box)
(9) If F5 contains the value 7 then F5 cannot contain the value 5 (the cell can contain only one value)
(10) If F6 contains the value 1 (6) then F6 cannot contain the value 5 (the cell can contain only one value)
(11) If F6 does not contain the value 5 and F5 does not contain the value 5 (9) then F8 must contain the value 5 (only remaining position in the column)
(12) If F8 contains the value 5 then G8 cannot contain the value 5 (the value can occur only once in the row)
(13) If F6 contains the value 1 (6) then D4 cannot contain the value 1 (the value can occur only once in the box)
(14) If E1 contains the value 1 (2) then E4 cannot contain the value 1 (the value can occur only once in the col)
(15) If A3 contains the value 1 (4) then I3 cannot contain the value 1 (the value can occur only once in the row)
(16) If A3 contains the value 1 (4) then G3 cannot contain the value 1 (the value can occur only once in the row)
(17) If G3 does not contain the value 1 and I3 does not contain the value 1 (15) then I2 must contain the value 1 (only remaining position in the box)
(18) If I2 contains the value 1 then I4 cannot contain the value 1 (the value can occur only once in the col)
(19) If I4 does not contain the value 1, E4 does not contain the value 1 (14) and D4 does not contain the value 1 (13) then G4 must contain the value 1 (only remaining position in the row)
(20) If G4 contains the value 1 then G4 cannot contain the value 5 (the cell can contain only one value)
(21) If G4 does not contain the value 5 and G8 does not contain the value 5 (12) then G6 must contain the value 5 (only remaining position in the column)
(22) If G6 contains the value 5 then G6 cannot contain the value 4 (the cell can contain only one value)
(23) If G4 contains the value 1 (19) then G4 cannot contain the value 4 (the cell can contain only one value)
(24) If A3 contains the value 1 (4) then A3 cannot contain the value 6 (the cell can contain only one value)
(25) If A3 does not contain the value 6 then C3 must contain the value 6 (only remaining position in the box)
(26) If C3 contains the value 6 then C9 cannot contain the value 6 (the value can occur only once in the col)
(27) If C9 does not contain the value 6 then B9 must contain the value 6 (only remaining position in the box)
(28) If B9 contains the value 6 then B4 cannot contain the value 6 (the value can occur only once in the col)
(29) If B4 does not contain the value 6 then B4 must contain the value 4 (only remaining potential value in the cell)
(30) If B4 contains the value 4 then H4 cannot contain the value 4 (the value can occur only once in the row)
(31) If B4 contains the value 4 (29) then I4 cannot contain the value 4 (the value can occur only once in the row)
(32) If B9 contains the value 6 (27) then B9 cannot contain the value 2 (the cell can contain only one value)
(33) If B9 does not contain the value 2 then G9 must contain the value 2 (only remaining position in the row)
(34) If G9 contains the value 2 then H7 cannot contain the value 2 (the value can occur only once in the box)
(35) If F6 does not contain the value 7 (7) then H6 must contain the value 7 (only remaining position in the row)
(36) If H6 contains the value 7 then H6 cannot contain the value 2 (the cell can contain only one value)
(37) If H6 does not contain the value 2 and H7 does not contain the value 2 (34) then H5 must contain the value 2 (only remaining position in the column)
(38) If H5 contains the value 2 then H5 cannot contain the value 4 (the cell can contain only one value)
(39) If B4 contains the value 4 (29) then D4 cannot contain the value 4 (the value can occur only once in the row)
(40) If E1 contains the value 1 (2) then E7 cannot contain the value 1 (the value can occur only once in the col)
(41) If E7 does not contain the value 1 then D7 must contain the value 1 (only remaining position in the box)
(42) If D7 contains the value 1 then D7 cannot contain the value 4 (the cell can contain only one value)
(43) If D7 contains the value 1 (41) then D7 cannot contain the value 2 (the cell can contain only one value)
(44) If D7 does not contain the value 2 then D8 must contain the value 2 (only remaining position in the box)
(45) If D8 contains the value 2 then D8 cannot contain the value 4 (the cell can contain only one value)
(46) If D8 does not contain the value 4, D7 does not contain the value 4 (42) and D4 does not contain the value 4 (39) then D5 must contain the value 4 (only remaining position in the column)
(47) If D5 contains the value 4 then I5 cannot contain the value 4 (the value can occur only once in the row)
(48) If H6 contains the value 7 (35) then H6 cannot contain the value 4 (the cell can contain only one value)
(49) If H6 does not contain the value 4, I5 does not contain the value 4 (47), H5 does not contain the value 4 (38), I4 does not contain the value 4 (31), H4 does not contain the value 4 (30), G4 does not contain the value 4 (23) and G6 does not contain the value 4 (22) then I6 must contain the value 4 (only remaining position in the box)
(50) If I6 contains the value 4 then I9 cannot contain the value 4 (the value can occur only once in the col)