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 does not contain the value 1, then I9 must contain the value 4 (View 1):
(1) If A1 does not contain the value 1 then A3 must contain the value 1 (only remaining position in the box)
(2) If A3 contains the value 1 then A3 cannot contain the value 6 (the cell can contain only one value)
(3) If A3 does not contain the value 6 then C3 must contain the value 6 (only remaining position in the box)
(4) If C3 contains the value 6 then C9 cannot contain the value 6 (the value can occur only once in the col)
(5) If C9 does not contain the value 6 then B9 must contain the value 6 (only remaining position in the box)
(6) If B9 contains the value 6 then B4 cannot contain the value 6 (the value can occur only once in the col)
(7) If B4 does not contain the value 6 then B4 must contain the value 4 (only remaining potential value in the cell)
(8) If B4 contains the value 4 then D4 cannot contain the value 4 (the value can occur only once in the row)
(9) If A1 does not contain the value 1 (initial assumption) then E1 must contain the value 1 (only remaining position in the row)
(10) If E1 contains the value 1 then F2 cannot contain the value 1 (the value can occur only once in the box)
(11) If A3 contains the value 1 (misplaced towel) then F3 cannot contain the value 1 (the value can occur only once in the row)
(12) If F3 does not contain the value 1 and F2 does not contain the value 1 (10) then F6 must contain the value 1 (only remaining position in the column)
(13) If F6 contains the value 1 then D4 cannot contain the value 1 (the value can occur only once in the box)
(14) If D4 does not contain the value 1 and D4 does not contain the value 4 (8) then D4 must contain the value 8 (only remaining potential value in the cell)
(15) If D4 contains the value 8 then E5 cannot contain the value 8 (the value can occur only once in the box)
(16) If D4 contains the value 8 (14) then D5 cannot contain the value 8 (the value can occur only once in the box)
(17) If F6 contains the value 1 (12) then F6 cannot contain the value 7 (the cell can contain only one value)
(18) If F6 does not contain the value 7 then F5 must contain the value 7 (only remaining position in the box)
(19) If F5 contains the value 7 then F5 cannot contain the value 8 (the cell can contain only one value)
(20) If B9 contains the value 6 (5) then B9 cannot contain the value 2 (the cell can contain only one value)
(21) If B9 does not contain the value 2 then G9 must contain the value 2 (only remaining position in the row)
(22) If G9 contains the value 2 then H7 cannot contain the value 2 (the value can occur only once in the box)
(23) If F6 does not contain the value 7 (17) then H6 must contain the value 7 (only remaining position in the row)
(24) If H6 contains the value 7 then H6 cannot contain the value 2 (the cell can contain only one value)
(25) If H6 does not contain the value 2 and H7 does not contain the value 2 (22) then H5 must contain the value 2 (only remaining position in the column)
(26) If H5 contains the value 2 then H5 cannot contain the value 8 (the cell can contain only one value)
(27) If H5 does not contain the value 8, F5 does not contain the value 8 (19), D5 does not contain the value 8 (16) and E5 does not contain the value 8 (15) then I5 must contain the value 8 (only remaining position in the row)
(28) If I5 contains the value 8 then I9 cannot contain the value 8 (the value can occur only once in the col)
(29) If F5 contains the value 7 (18) then F5 cannot contain the value 5 (the cell can contain only one value)
(30) If F6 contains the value 1 (12) then F6 cannot contain the value 5 (the cell can contain only one value)
(31) If F6 does not contain the value 5 and F5 does not contain the value 5 (29) then F8 must contain the value 5 (only remaining position in the column)
(32) If F8 contains the value 5 then F8 cannot contain the value 9 (the cell can contain only one value)
(33) If F8 does not contain the value 9 then F9 must contain the value 9 (only remaining position in the box)
(34) If F9 contains the value 9 then I9 cannot contain the value 9 (the value can occur only once in the row)
(35) If I9 does not contain the value 9 and I9 does not contain the value 8 (28) then I9 must contain the value 4 (only remaining potential value in the cell)
Chain 2: If A1 does not contain the value 1, then I9 cannot contain the value 4 (View 2):
(1) If A1 does not contain the value 1 then E1 must contain the value 1 (only remaining position in the row)
(2) If E1 contains the value 1 then F2 cannot contain the value 1 (the value can occur only once in the box)
(3) If A1 does not contain the value 1 (initial assumption) then A3 must contain the value 1 (only remaining position in the box)
(4) If A3 contains the value 1 then F3 cannot contain the value 1 (the value can occur only once in the row)
(5) If F3 does not contain the value 1 and F2 does not contain the value 1 (2) then F6 must contain the value 1 (only remaining position in the column)
(6) If F6 contains the value 1 then F6 cannot contain the value 7 (the cell can contain only one value)
(7) If F6 does not contain the value 7 then F5 must contain the value 7 (only remaining position in the box)
(8) If F5 contains the value 7 then F5 cannot contain the value 5 (the cell can contain only one value)
(9) If F6 contains the value 1 (5) then F6 cannot contain the value 5 (the cell can contain only one value)
(10) If F6 does not contain the value 5 and F5 does not contain the value 5 (8) then F8 must contain the value 5 (only remaining position in the column)
(11) If F8 contains the value 5 then G8 cannot contain the value 5 (the value can occur only once in the row)
(12) If F6 contains the value 1 (5) then D4 cannot contain the value 1 (the value can occur only once in the box)
(13) If E1 contains the value 1 (misplaced towel) then E4 cannot contain the value 1 (the value can occur only once in the col)
(14) If A3 contains the value 1 (3) then I3 cannot contain the value 1 (the value can occur only once in the row)
(15) If A3 contains the value 1 (3) then G3 cannot contain the value 1 (the value can occur only once in the row)
(16) If G3 does not contain the value 1 and I3 does not contain the value 1 (14) then I2 must contain the value 1 (only remaining position in the box)
(17) If I2 contains the value 1 then I4 cannot contain the value 1 (the value can occur only once in the col)
(18) If I4 does not contain the value 1, E4 does not contain the value 1 (13) and D4 does not contain the value 1 (12) then G4 must contain the value 1 (only remaining position in the row)
(19) If G4 contains the value 1 then G4 cannot contain the value 5 (the cell can contain only one value)
(20) If G4 does not contain the value 5 and G8 does not contain the value 5 (11) then G6 must contain the value 5 (only remaining position in the column)
(21) If G6 contains the value 5 then G6 cannot contain the value 4 (the cell can contain only one value)
(22) If G4 contains the value 1 (18) then G4 cannot contain the value 4 (the cell can contain only one value)
(23) If A3 contains the value 1 (3) then A3 cannot contain the value 6 (the cell can contain only one value)
(24) If A3 does not contain the value 6 then C3 must contain the value 6 (only remaining position in the box)
(25) If C3 contains the value 6 then C9 cannot contain the value 6 (the value can occur only once in the col)
(26) If C9 does not contain the value 6 then B9 must contain the value 6 (only remaining position in the box)
(27) If B9 contains the value 6 then B4 cannot contain the value 6 (the value can occur only once in the col)
(28) If B4 does not contain the value 6 then B4 must contain the value 4 (only remaining potential value in the cell)
(29) If B4 contains the value 4 then H4 cannot contain the value 4 (the value can occur only once in the row)
(30) If B4 contains the value 4 (28) then I4 cannot contain the value 4 (the value can occur only once in the row)
(31) If B9 contains the value 6 (26) then B9 cannot contain the value 2 (the cell can contain only one value)
(32) If B9 does not contain the value 2 then G9 must contain the value 2 (only remaining position in the row)
(33) If G9 contains the value 2 then H7 cannot contain the value 2 (the value can occur only once in the box)
(34) If F6 does not contain the value 7 (6) then H6 must contain the value 7 (only remaining position in the row)
(35) If H6 contains the value 7 then H6 cannot contain the value 2 (the cell can contain only one value)
(36) If H6 does not contain the value 2 and H7 does not contain the value 2 (33) then H5 must contain the value 2 (only remaining position in the column)
(37) If H5 contains the value 2 then H5 cannot contain the value 4 (the cell can contain only one value)
(38) If B4 contains the value 4 (28) then D4 cannot contain the value 4 (the value can occur only once in the row)
(39) If E1 contains the value 1 (misplaced towel) then E7 cannot contain the value 1 (the value can occur only once in the col)
(40) If E7 does not contain the value 1 then D7 must contain the value 1 (only remaining position in the box)
(41) If D7 contains the value 1 then D7 cannot contain the value 4 (the cell can contain only one value)
(42) If D7 contains the value 1 (40) then D7 cannot contain the value 2 (the cell can contain only one value)
(43) If D7 does not contain the value 2 then D8 must contain the value 2 (only remaining position in the box)
(44) If D8 contains the value 2 then D8 cannot contain the value 4 (the cell can contain only one value)
(45) If D8 does not contain the value 4, D7 does not contain the value 4 (41) and D4 does not contain the value 4 (38) then D5 must contain the value 4 (only remaining position in the column)
(46) If D5 contains the value 4 then I5 cannot contain the value 4 (the value can occur only once in the row)
(47) If H6 contains the value 7 (34) then H6 cannot contain the value 4 (the cell can contain only one value)
(48) If H6 does not contain the value 4, I5 does not contain the value 4 (46), H5 does not contain the value 4 (37), I4 does not contain the value 4 (30), H4 does not contain the value 4 (29), G4 does not contain the value 4 (22) and G6 does not contain the value 4 (21) then I6 must contain the value 4 (only remaining position in the box)
(49) If I6 contains the value 4 then I9 cannot contain the value 4 (the value can occur only once in the col)