Dynamic Binary Value Contradiction Forcing Chain

With this solving technique, we will prove the two following assertions:

Because the same assumption yields two contradictory results, we can conclude that the assumption is false, that is, D2 cannot contain the value 6.

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 D2 contains the value 6, then I4 must contain the value 7 (View 1):
(1) If D2 contains the value 6 then G2 cannot contain the value 6 (the value can occur only once in the row)
(2) If G2 does not contain the value 6 then G3 must contain the value 6 (only remaining position in the box)
(3) If G3 contains the value 6 then G3 cannot contain the value 7 (the cell can contain only one value)
(4) If G3 does not contain the value 7 then H3 must contain the value 7 (only remaining position in the box)
(5) If H3 contains the value 7 then H3 cannot contain the value 2 (the cell can contain only one value)
(6) If G3 contains the value 6 (2) then G3 cannot contain the value 2 (the cell can contain only one value)
(7) If G3 does not contain the value 2 and H3 does not contain the value 2 (5) then D3 must contain the value 2 (only remaining position in the row)
(8) If D3 contains the value 2 then D3 cannot contain the value 4 (the cell can contain only one value)
(9) If D2 contains the value 6 (initial assumption) then D2 cannot contain the value 4 (the cell can contain only one value)
(10) If D2 does not contain the value 4 and D3 does not contain the value 4 (8) then F2 must contain the value 4 (only remaining position in the box)
(11) If F2 contains the value 4 then C2 cannot contain the value 4 (the value can occur only once in the row)
(12) If D2 contains the value 6 (initial assumption) then C2 cannot contain the value 6 (the value can occur only once in the row)
(13) If C2 does not contain the value 6 and C2 does not contain the value 4 (11) then C2 must contain the value 5 (only remaining potential value in the cell)
(14) If C2 contains the value 5 then B3 cannot contain the value 5 (the value can occur only once in the box)
(15) If C2 contains the value 5 (13) then A3 cannot contain the value 5 (the value can occur only once in the box)
(16) If G3 contains the value 6 (2) then G3 cannot contain the value 5 (the cell can contain only one value)
(17) If H3 contains the value 7 (4) then H3 cannot contain the value 5 (the cell can contain only one value)
(18) If H3 does not contain the value 5, G3 does not contain the value 5 (16), A3 does not contain the value 5 (15) and B3 does not contain the value 5 (14) then E3 must contain the value 5 (only remaining position in the row)
(19) If E3 contains the value 5 then F1 cannot contain the value 5 (the value can occur only once in the box)
(20) If D3 contains the value 2 (7) then F1 cannot contain the value 2 (the value can occur only once in the box)
(21) If F2 contains the value 4 (10) then F5 cannot contain the value 4 (the value can occur only once in the col)
(22) If F5 does not contain the value 4 then F5 must contain the value 3 (only remaining potential value in the cell)
(23) If F5 contains the value 3 then F1 cannot contain the value 3 (the value can occur only once in the col)
(24) If D2 contains the value 6 (initial assumption) then F1 cannot contain the value 6 (the value can occur only once in the box)
(25) If F1 does not contain the value 6, F1 does not contain the value 3 (23), F1 does not contain the value 2 (20) and F1 does not contain the value 5 (19) then F1 must contain the value 9 (only remaining potential value in the cell)
(26) If F1 contains the value 9 then F8 cannot contain the value 9 (the value can occur only once in the col)
(27) If F8 does not contain the value 9 then D8 must contain the value 9 (only remaining position in the box)
(28) If D8 contains the value 9 then D8 cannot contain the value 7 (the cell can contain only one value)
(29) If D8 does not contain the value 7 then D9 must contain the value 7 (only remaining position in the box)
(30) If D9 contains the value 7 then B9 cannot contain the value 7 (the value can occur only once in the row)
(31) If D2 contains the value 6 (initial assumption) then D1 cannot contain the value 6 (the value can occur only once in the box)
(32) If D2 contains the value 6 (initial assumption) then E1 cannot contain the value 6 (the value can occur only once in the box)
(33) If E1 does not contain the value 6, D1 does not contain the value 6 (31) and F1 does not contain the value 6 (24) then C1 must contain the value 6 (only remaining position in the row)
(34) If C1 contains the value 6 then C1 cannot contain the value 8 (the cell can contain only one value)
(35) If C1 does not contain the value 8 then B1 must contain the value 8 (only remaining position in the box)
(36) If B1 contains the value 8 then B9 cannot contain the value 8 (the value can occur only once in the col)
(37) If B9 does not contain the value 8 and B9 does not contain the value 7 (30) then B9 must contain the value 4 (only remaining potential value in the cell)
(38) If B9 contains the value 4 then B6 cannot contain the value 4 (the value can occur only once in the col)
(39) If B1 contains the value 8 (35) then B6 cannot contain the value 8 (the value can occur only once in the col)
(40) If F5 contains the value 3 (22) then E6 cannot contain the value 3 (the value can occur only once in the box)
(41) If E6 does not contain the value 3 then E1 must contain the value 3 (only remaining position in the column)
(42) If E1 contains the value 3 then E1 cannot contain the value 1 (the cell can contain only one value)
(43) If E3 contains the value 5 (18) then E3 cannot contain the value 1 (the cell can contain only one value)
(44) If E3 does not contain the value 1 and E1 does not contain the value 1 (42) then E4 must contain the value 1 (only remaining position in the column)
(45) If E4 contains the value 1 then E4 cannot contain the value 9 (the cell can contain only one value)
(46) If B1 contains the value 8 (35) then B1 cannot contain the value 9 (the cell can contain only one value)
(47) If B1 does not contain the value 9 then A2 must contain the value 9 (only remaining position in the box)
(48) If A2 contains the value 9 then A4 cannot contain the value 9 (the value can occur only once in the col)
(49) If F1 contains the value 9 (25) then F4 cannot contain the value 9 (the value can occur only once in the col)
(50) If F4 does not contain the value 9, A4 does not contain the value 9 (48) and E4 does not contain the value 9 (45) then B4 must contain the value 9 (only remaining position in the row)
(51) If B4 contains the value 9 then B6 cannot contain the value 9 (the value can occur only once in the box)
(52) If B6 does not contain the value 9, B6 does not contain the value 8 (39) and B6 does not contain the value 4 (38) then B6 must contain the value 5 (only remaining potential value in the cell)
(53) If B6 contains the value 5 then A5 cannot contain the value 5 (the value can occur only once in the box)
(54) If C2 contains the value 5 (13) then C5 cannot contain the value 5 (the value can occur only once in the col)
(55) If C5 does not contain the value 5 and A5 does not contain the value 5 (53) then G5 must contain the value 5 (only remaining position in the row)
(56) If G5 contains the value 5 then G5 cannot contain the value 7 (the cell can contain only one value)
(57) If H3 contains the value 7 (4) then H4 cannot contain the value 7 (the value can occur only once in the col)
(58) If D2 contains the value 6 (initial assumption) then D2 cannot contain the value 1 (the cell can contain only one value)
(59) If D3 contains the value 2 (7) then D3 cannot contain the value 1 (the cell can contain only one value)
(60) If E3 does not contain the value 1 (43), D3 does not contain the value 1, D2 does not contain the value 1 (58) and E1 does not contain the value 1 (42) then D1 must contain the value 1 (only remaining position in the box)
(61) If D1 contains the value 1 then D5 cannot contain the value 1 (the value can occur only once in the col)
(62) If F2 contains the value 4 (10) then F2 cannot contain the value 3 (the cell can contain only one value)
(63) If F2 does not contain the value 3 then I2 must contain the value 3 (only remaining position in the row)
(64) If I2 contains the value 3 then I2 cannot contain the value 1 (the cell can contain only one value)
(65) If A2 contains the value 9 (47) then A2 cannot contain the value 1 (the cell can contain only one value)
(66) If D2 does not contain the value 1 (58), A2 does not contain the value 1 and I2 does not contain the value 1 (64) then G2 must contain the value 1 (only remaining position in the row)
(67) If G2 contains the value 1 then G5 cannot contain the value 1 (the value can occur only once in the col)
(68) If G5 does not contain the value 1 and D5 does not contain the value 1 (61) then I5 must contain the value 1 (only remaining position in the row)
(69) If I5 contains the value 1 then I5 cannot contain the value 7 (the cell can contain only one value)
(70) If I5 does not contain the value 7, H4 does not contain the value 7 (57) and G5 does not contain the value 7 (56) then I4 must contain the value 7 (only remaining position in the box)

Chain 2: If D2 contains the value 6, then I4 cannot contain the value 7 (View 2):
(1) If D2 contains the value 6 then G2 cannot contain the value 6 (the value can occur only once in the row)
(2) If G2 does not contain the value 6 then G3 must contain the value 6 (only remaining position in the box)
(3) If G3 contains the value 6 then G3 cannot contain the value 7 (the cell can contain only one value)
(4) If G3 does not contain the value 7 then H3 must contain the value 7 (only remaining position in the box)
(5) If H3 contains the value 7 then H3 cannot contain the value 2 (the cell can contain only one value)
(6) If G3 contains the value 6 (2) then G3 cannot contain the value 2 (the cell can contain only one value)
(7) If G3 does not contain the value 2 and H3 does not contain the value 2 (5) then D3 must contain the value 2 (only remaining position in the row)
(8) If D3 contains the value 2 then D3 cannot contain the value 4 (the cell can contain only one value)
(9) If D2 contains the value 6 (initial assumption) then D2 cannot contain the value 4 (the cell can contain only one value)
(10) If D2 does not contain the value 4 and D3 does not contain the value 4 (8) then F2 must contain the value 4 (only remaining position in the box)
(11) If F2 contains the value 4 then C2 cannot contain the value 4 (the value can occur only once in the row)
(12) If D2 contains the value 6 (initial assumption) then C2 cannot contain the value 6 (the value can occur only once in the row)
(13) If C2 does not contain the value 6 and C2 does not contain the value 4 (11) then C2 must contain the value 5 (only remaining potential value in the cell)
(14) If C2 contains the value 5 then B3 cannot contain the value 5 (the value can occur only once in the box)
(15) If C2 contains the value 5 (13) then A3 cannot contain the value 5 (the value can occur only once in the box)
(16) If G3 contains the value 6 (2) then G3 cannot contain the value 5 (the cell can contain only one value)
(17) If H3 contains the value 7 (4) then H3 cannot contain the value 5 (the cell can contain only one value)
(18) If H3 does not contain the value 5, G3 does not contain the value 5 (16), A3 does not contain the value 5 (15) and B3 does not contain the value 5 (14) then E3 must contain the value 5 (only remaining position in the row)
(19) If E3 contains the value 5 then F1 cannot contain the value 5 (the value can occur only once in the box)
(20) If D3 contains the value 2 (7) then F1 cannot contain the value 2 (the value can occur only once in the box)
(21) If F2 contains the value 4 (10) then F5 cannot contain the value 4 (the value can occur only once in the col)
(22) If F5 does not contain the value 4 then F5 must contain the value 3 (only remaining potential value in the cell)
(23) If F5 contains the value 3 then F1 cannot contain the value 3 (the value can occur only once in the col)
(24) If D2 contains the value 6 (initial assumption) then F1 cannot contain the value 6 (the value can occur only once in the box)
(25) If F1 does not contain the value 6, F1 does not contain the value 3 (23), F1 does not contain the value 2 (20) and F1 does not contain the value 5 (19) then F1 must contain the value 9 (only remaining potential value in the cell)
(26) If F1 contains the value 9 then F8 cannot contain the value 9 (the value can occur only once in the col)
(27) If F8 does not contain the value 9 then D8 must contain the value 9 (only remaining position in the box)
(28) If D8 contains the value 9 then D8 cannot contain the value 7 (the cell can contain only one value)
(29) If D8 does not contain the value 7 then D9 must contain the value 7 (only remaining position in the box)
(30) If D9 contains the value 7 then B9 cannot contain the value 7 (the value can occur only once in the row)
(31) If D2 contains the value 6 (initial assumption) then D1 cannot contain the value 6 (the value can occur only once in the box)
(32) If D2 contains the value 6 (initial assumption) then E1 cannot contain the value 6 (the value can occur only once in the box)
(33) If E1 does not contain the value 6, D1 does not contain the value 6 (31) and F1 does not contain the value 6 (24) then C1 must contain the value 6 (only remaining position in the row)
(34) If C1 contains the value 6 then C1 cannot contain the value 8 (the cell can contain only one value)
(35) If C1 does not contain the value 8 then B1 must contain the value 8 (only remaining position in the box)
(36) If B1 contains the value 8 then B9 cannot contain the value 8 (the value can occur only once in the col)
(37) If B9 does not contain the value 8 and B9 does not contain the value 7 (30) then B9 must contain the value 4 (only remaining potential value in the cell)
(38) If B9 contains the value 4 then B6 cannot contain the value 4 (the value can occur only once in the col)
(39) If B1 contains the value 8 (35) then B6 cannot contain the value 8 (the value can occur only once in the col)
(40) If F5 contains the value 3 (22) then E6 cannot contain the value 3 (the value can occur only once in the box)
(41) If E6 does not contain the value 3 then E1 must contain the value 3 (only remaining position in the column)
(42) If E1 contains the value 3 then E1 cannot contain the value 1 (the cell can contain only one value)
(43) If E3 contains the value 5 (18) then E3 cannot contain the value 1 (the cell can contain only one value)
(44) If E3 does not contain the value 1 and E1 does not contain the value 1 (42) then E4 must contain the value 1 (only remaining position in the column)
(45) If E4 contains the value 1 then E4 cannot contain the value 9 (the cell can contain only one value)
(46) If B1 contains the value 8 (35) then B1 cannot contain the value 9 (the cell can contain only one value)
(47) If B1 does not contain the value 9 then A2 must contain the value 9 (only remaining position in the box)
(48) If A2 contains the value 9 then A4 cannot contain the value 9 (the value can occur only once in the col)
(49) If F1 contains the value 9 (25) then F4 cannot contain the value 9 (the value can occur only once in the col)
(50) If F4 does not contain the value 9, A4 does not contain the value 9 (48) and E4 does not contain the value 9 (45) then B4 must contain the value 9 (only remaining position in the row)
(51) If B4 contains the value 9 then B6 cannot contain the value 9 (the value can occur only once in the box)
(52) If B6 does not contain the value 9, B6 does not contain the value 8 (39) and B6 does not contain the value 4 (38) then B6 must contain the value 5 (only remaining potential value in the cell)
(53) If B6 contains the value 5 then A5 cannot contain the value 5 (the value can occur only once in the box)
(54) If F5 contains the value 3 (22) then A5 cannot contain the value 3 (the value can occur only once in the row)
(55) If D2 contains the value 6 (initial assumption) then D2 cannot contain the value 1 (the cell can contain only one value)
(56) If D3 contains the value 2 (7) then D3 cannot contain the value 1 (the cell can contain only one value)
(57) If E3 does not contain the value 1 (43), D3 does not contain the value 1, D2 does not contain the value 1 (55) and E1 does not contain the value 1 (42) then D1 must contain the value 1 (only remaining position in the box)
(58) If D1 contains the value 1 then D5 cannot contain the value 1 (the value can occur only once in the col)
(59) If F2 contains the value 4 (10) then F2 cannot contain the value 3 (the cell can contain only one value)
(60) If F2 does not contain the value 3 then I2 must contain the value 3 (only remaining position in the row)
(61) If I2 contains the value 3 then I2 cannot contain the value 1 (the cell can contain only one value)
(62) If A2 contains the value 9 (47) then A2 cannot contain the value 1 (the cell can contain only one value)
(63) If D2 does not contain the value 1 (55), A2 does not contain the value 1 and I2 does not contain the value 1 (61) then G2 must contain the value 1 (only remaining position in the row)
(64) If G2 contains the value 1 then G5 cannot contain the value 1 (the value can occur only once in the col)
(65) If G5 does not contain the value 1 and D5 does not contain the value 1 (58) then I5 must contain the value 1 (only remaining position in the row)
(66) If I5 contains the value 1 then I8 cannot contain the value 1 (the value can occur only once in the col)
(67) If G2 contains the value 1 (63) then G8 cannot contain the value 1 (the value can occur only once in the col)
(68) If G8 does not contain the value 1 and I8 does not contain the value 1 (66) then A8 must contain the value 1 (only remaining position in the row)
(69) If A8 contains the value 1 then A3 cannot contain the value 1 (the value can occur only once in the col)
(70) If C1 contains the value 6 (33) then A3 cannot contain the value 6 (the value can occur only once in the box)
(71) If A3 does not contain the value 6, A3 does not contain the value 5 (15) and A3 does not contain the value 1 (69) then A3 must contain the value 4 (only remaining potential value in the cell)
(72) If A3 contains the value 4 then A5 cannot contain the value 4 (the value can occur only once in the col)
(73) If A5 does not contain the value 4, A5 does not contain the value 3 (54) and A5 does not contain the value 5 (53) then A5 must contain the value 8 (only remaining potential value in the cell)
(74) If A5 contains the value 8 then C4 cannot contain the value 8 (the value can occur only once in the box)
(75) If A5 contains the value 8 (73) then A4 cannot contain the value 8 (the value can occur only once in the box)
(76) If B1 contains the value 8 (35) then B4 cannot contain the value 8 (the value can occur only once in the col)
(77) If E4 contains the value 1 (44) then E4 cannot contain the value 8 (the cell can contain only one value)
(78) If E4 does not contain the value 8, B4 does not contain the value 8 (76), A4 does not contain the value 8 (75) and C4 does not contain the value 8 (74) then I4 must contain the value 8 (only remaining position in the row)
(79) If I4 contains the value 8 then I4 cannot contain the value 7 (the cell can contain only one value)