The cells C4, A6, A5, A8, A4, A2, B3, C2, G2, G6 and I4 form a bidirectional cycle: there are exactly two ways of placing the values in these cells, forming two different possible configurations. Some values appear in a row, column or box regardless of which configuration is correct. Because one of the two configurations must be correct, other occurrences of these values can be removed from these rows, columns or boxs.
The cycle is an Y-Cycle because only cells with two potential values are involved. Bidirectional Y-Cycles are also known as Forcing Loops or Bivalue cycles.
The two configurations of the bidirectional cycle are enforced by two complementary, circular chains of implications. The details of each chain are given below.
Cycle details, forward direction (View 1):
(1) If I4 contains the value 4 then C4 cannot contain the value 4 (the value can occur only once in the row)
(2) If C4 does not contain the value 4 then C4 must contain the value 2 (only remaining potential value in the cell)
(3) If C4 contains the value 2 then A6 cannot contain the value 2 (the value can occur only once in the box)
(4) If A6 does not contain the value 2 then A6 must contain the value 3 (only remaining potential value in the cell)
(5) If A6 contains the value 3 then A5 cannot contain the value 3 (the value can occur only once in the box)
(6) If A5 does not contain the value 3 then A5 must contain the value 5 (only remaining potential value in the cell)
(7) If A5 contains the value 5 then A8 cannot contain the value 5 (the value can occur only once in the col)
(8) If A8 does not contain the value 5 then A8 must contain the value 8 (only remaining potential value in the cell)
(9) If A8 contains the value 8 then A4 cannot contain the value 8 (the value can occur only once in the col)
(10) If A4 does not contain the value 8 then A4 must contain the value 9 (only remaining potential value in the cell)
(11) If A4 contains the value 9 then A2 cannot contain the value 9 (the value can occur only once in the col)
(12) If A2 does not contain the value 9 then A2 must contain the value 1 (only remaining potential value in the cell)
(13) If A2 contains the value 1 then B3 cannot contain the value 1 (the value can occur only once in the box)
(14) If B3 does not contain the value 1 then B3 must contain the value 5 (only remaining potential value in the cell)
(15) If B3 contains the value 5 then C2 cannot contain the value 5 (the value can occur only once in the box)
(16) If C2 does not contain the value 5 then C2 must contain the value 6 (only remaining potential value in the cell)
(17) If C2 contains the value 6 then G2 cannot contain the value 6 (the value can occur only once in the row)
(18) If G2 does not contain the value 6 then G2 must contain the value 5 (only remaining potential value in the cell)
(19) If G2 contains the value 5 then G6 cannot contain the value 5 (the value can occur only once in the col)
(20) If G6 does not contain the value 5 then G6 must contain the value 2 (only remaining potential value in the cell)
(21) If G6 contains the value 2 then I4 cannot contain the value 2 (the value can occur only once in the box)
(22) If I4 does not contain the value 2 then I4 must contain the value 4 (only remaining potential value in the cell)
Cycle details, reverse direction (View 2):
(1) If I4 does not contain the value 4 then I4 must contain the value 2 (only remaining potential value in the cell)
(2) If I4 contains the value 2 then G6 cannot contain the value 2 (the value can occur only once in the box)
(3) If G6 does not contain the value 2 then G6 must contain the value 5 (only remaining potential value in the cell)
(4) If G6 contains the value 5 then G2 cannot contain the value 5 (the value can occur only once in the col)
(5) If G2 does not contain the value 5 then G2 must contain the value 6 (only remaining potential value in the cell)
(6) If G2 contains the value 6 then C2 cannot contain the value 6 (the value can occur only once in the row)
(7) If C2 does not contain the value 6 then C2 must contain the value 5 (only remaining potential value in the cell)
(8) If C2 contains the value 5 then B3 cannot contain the value 5 (the value can occur only once in the box)
(9) If B3 does not contain the value 5 then B3 must contain the value 1 (only remaining potential value in the cell)
(10) If B3 contains the value 1 then A2 cannot contain the value 1 (the value can occur only once in the box)
(11) If A2 does not contain the value 1 then A2 must contain the value 9 (only remaining potential value in the cell)
(12) If A2 contains the value 9 then A4 cannot contain the value 9 (the value can occur only once in the col)
(13) If A4 does not contain the value 9 then A4 must contain the value 8 (only remaining potential value in the cell)
(14) If A4 contains the value 8 then A8 cannot contain the value 8 (the value can occur only once in the col)
(15) If A8 does not contain the value 8 then A8 must contain the value 5 (only remaining potential value in the cell)
(16) If A8 contains the value 5 then A5 cannot contain the value 5 (the value can occur only once in the col)
(17) If A5 does not contain the value 5 then A5 must contain the value 3 (only remaining potential value in the cell)
(18) If A5 contains the value 3 then A6 cannot contain the value 3 (the value can occur only once in the box)
(19) If A6 does not contain the value 3 then A6 must contain the value 2 (only remaining potential value in the cell)
(20) If A6 contains the value 2 then C4 cannot contain the value 2 (the value can occur only once in the box)
(21) If C4 does not contain the value 2 then C4 must contain the value 4 (only remaining potential value in the cell)
(22) If C4 contains the value 4 then I4 cannot contain the value 4 (the value can occur only once in the row)