Killer Sudoku Combinations

For a cage of k cells and target sum S, valid combinations are all strictly increasing k-tuples (d_1 < d_2 < ... < d_k) drawn from {1..9} satisfying d_1 + d_2 + ... + d_k = S, filtered by required inclusions I and peer exclusions E.

Plain-Text Partition & Set Intersection Formulas

• Feasible combination set: C(k, S, I, E) = { D ⊆ ({1..9} \ E) : |D| = k, ∑_{d ∈ D} d = S, I ⊆ D }
• Minimum & maximum sum bounds for k cells: S_min(k) = ∑_{i=1..k} i = k(k + 1)/2 and S_max(k) = ∑_{i=10-k..9} i = k(19 - k)/2
• Mandatory digits (intersection across all valid sets): Mandatory(k, S, I, E) = ⋂_{D ∈ C(k,S,I,E)} D
• Eliminated digits (complement of union): Eliminated(k, S, I, E) = {1..9} \ ⋃_{D ∈ C(k,S,I,E)} D

Worked Filtering Example With Real Numbers

Suppose a 3-cell cage has target sum S = 10 and lies entirely inside Row 4, where digit 1 is already placed outside the cage:

1. Unfiltered combinations for (k = 3, S = 10): Enumerating 3-subsets of {1..9} summing to 10 yields 4 combinations: [1, 2, 7], [1, 3, 6], [1, 4, 5], and [2, 3, 5]. Notice that 8 and 9 never appear (`Eliminated = {8, 9}`).
2. Applying exclusion E = {1}: Removing every combination containing 1 eliminates [1, 2, 7], [1, 3, 6], and [1, 4, 5].
3. Exact solved result: Only 1 combination remains: [2, 3, 5] (`2 + 3 + 5 = 10`). Therefore, Mandatory = {2, 3, 5} and Eliminated = {1, 4, 6, 7, 8, 9}—locking digits 2, 3, and 5 as a Naked Triple in Row 4.

Killer Sudoku Unique & High-Yield Cage Combinations Table (2 to 7 Cells)