Cycle Killer Redux

(Published yesterday, 22:52 by SirSchmoopy)

Rules:

Normal sudoku rules apply.

Normal killer sudoku rules apply. Digits in a cage cannot repeat and sum to the total indicated in the top left corner.

Let a cycle be a series of cells in a single row created by repeatedly jumping to the column indexed by the current cell's digit. (e.g. if a cell's digit was a 3, the next cell in the cycle would be the cell in column 3)

Let the order of a cycle be the number of unique digits in a cycle. e.g. - If a row in the grid read left to right was 234517698, then the 2 in column 1 would be part of a cycle of 2>3>4>5>1 with order 5. The 9 in column 8 and the 8 in column 9 would form and order 2 cycle

Along with regular killer sudoku rules, the orders of cycles in a cage do not repeat and sum to the value of the cage's clue. (Single-cell clueless cages have a digit equal to their order)

Example:

See below for a valid and invalid killer cage under this additional killer cycle constraint.

In the first (valid) example, the order of the 4 is 1 (4 is in column 4 so it points to itself). The 2 is in an order 5 cycle (2 -> 3 ->5 -> 7 -> 4) and 4+2=1+5=6

The second example is invalid for two reasons:

The 1 is in an order 3 cycle. This immediately makes the cage invalid as the order of the other cycle would have to be 3 in order to sum to 6, and orders cannot repeat within cages.
The 5 is in a cycle with an order of at least 4, so the sum of the orders in the cage is at least 3+4=7. To show the cycle is an order of at least 4, consider the following:
The 5 is in the 6th column, so the 5 and the 6 are part of the cycle.
The 6 is in the 7th column which means the 7 is part of the cycle.
The 7 is in the 3rd column, so the 3 in that row is part of the cycle as well.
The smallest cycle is then order 4 by placing a 3 in R6C5.