Bidirectional Y-Cycle

The cells A9, H9, B9, B1 and A1 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 A1 contains the value 5 then A9 cannot contain the value 5 (the value can occur only once in the col)
(2) If A9 does not contain the value 5 then A9 must contain the value 9 (only remaining potential value in the cell)
(3) If A9 contains the value 9 then H9 cannot contain the value 9 (the value can occur only once in the row)
(4) If H9 does not contain the value 9 then H9 must contain the value 8 (only remaining potential value in the cell)
(5) If H9 contains the value 8 then B9 cannot contain the value 8 (the value can occur only once in the row)
(6) If B9 does not contain the value 8 then B9 must contain the value 2 (only remaining potential value in the cell)
(7) If B9 contains the value 2 then B1 cannot contain the value 2 (the value can occur only once in the col)
(8) If B1 does not contain the value 2 then B1 must contain the value 9 (only remaining potential value in the cell)
(9) If B1 contains the value 9 then A1 cannot contain the value 9 (the value can occur only once in the box)
(10) If A1 does not contain the value 9 then A1 must contain the value 5 (only remaining potential value in the cell)

Cycle details, reverse direction (View 2):
(1) If A1 does not contain the value 5 then A1 must contain the value 9 (only remaining potential value in the cell)
(2) If A1 contains the value 9 then B1 cannot contain the value 9 (the value can occur only once in the box)
(3) If B1 does not contain the value 9 then B1 must contain the value 2 (only remaining potential value in the cell)
(4) If B1 contains the value 2 then B9 cannot contain the value 2 (the value can occur only once in the col)
(5) If B9 does not contain the value 2 then B9 must contain the value 8 (only remaining potential value in the cell)
(6) If B9 contains the value 8 then H9 cannot contain the value 8 (the value can occur only once in the row)
(7) If H9 does not contain the value 8 then H9 must contain the value 9 (only remaining potential value in the cell)
(8) If H9 contains the value 9 then A9 cannot contain the value 9 (the value can occur only once in the row)
(9) If A9 does not contain the value 9 then A9 must contain the value 5 (only remaining potential value in the cell)
(10) If A9 contains the value 5 then A1 cannot contain the value 5 (the value can occur only once in the col)