Lockout Line Sudoku Help

Lockout Line is drawn with two diamond (rotated square) cells at the ends. Digits on the line between the diamonds must not be between the digits in the two diamonds. In other words, every digit on the line must be lower than both diamond digits or higher than both diamond digits.

Additionally, the two diamond digits must differ by at least 4.

On the image above diamonds contain 1 and 7 (difference is 6). Line digits 8 and 9 are both higher than 7, so they are not between 1 and 7.

If only one diamond digit is set, we can limit candidates for the other diamond and for the line cells.

On the image above one diamond is 2. Since diamond digits must differ by at least 4, the other diamond can be only 6, 7, 8 or 9. If the other diamond digit is greater than 2, then the line digits must be lower than 2 (so only 1) or higher than the other diamond digit. That is why the line cell can be only 1, 7, 8 or 9. Digits 2, 3, 4, 5 and 6 cannot be on the line.

When several line cells are in the same house (row, column or box), those cells must have unique digits. All of them must be outside the interval between the two diamond digits. For diamond digits that differ by d, there are 8 - d digits outside that interval. This means that a line with N cells in a single house requires d >= 4 and d <= 8 - N.

The line on the image above has four cells in the same box, so N = 4. Because diamond digits must differ by at least 4 and at most 8 - 4 = 4, they differ by exactly 4. Since one diamond contains 1, the other diamond must be 5, and the four line cells must use 6, 7, 8 and 9.

The image above shows the same line with both diamonds set.