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The solver combines ordinary Sudoku restrictions with cage sums and non-repeating digits.

Cage Arithmetic Formula & Worked Example

How does the Killer Sudoku solver combine cage sums with 9×9 Sudoku rules?

The Killer Sudoku Solver enforces standard 9×9 row, column, and 3×3 box uniqueness alongside two cage equations for every dotted cage C_k with target sum S_k: the digits inside C_k must sum to S_k, no digit may repeat inside C_k, and all 81 grid cells must partition into cages whose total sum equals 405.

Plain-Text Cage & Rule-of-45 Equations

For each cage C_k ⊆ {1..9}² of size n_k = |C_k| and target sum S_k, let v(r, c) ∈ {1..9} be the digit placed in cell (r, c):

Cage sum equation: ∑_{(r,c) ∈ C_k} v(r, c) = S_k
Cage non-repetition rule: ∀ (r_1, c_1) ≠ (r_2, c_2) ∈ C_k: v(r_1, c_1) ≠ v(r_2, c_2)
Sum feasibility bounds for an n-cell cage: S_min(n) = n(n + 1)/2 and S_max(n) = n(19 - n)/2
Rule of 45 (Innie / Outie equation): Because every row, column, or 3×3 box U satisfies ∑_{(r,c) ∈ U} v(r, c) = 45, any k-unit region R has total sum 45k. If cages inside U cover 8 of 9 cells with sum ∑ S_inside, the remaining 9th cell (“innie”) equals v_innie = 45 - ∑ S_inside.

Worked Killer Sudoku Deduction Example With Real Numbers

Consider Box 1 (rows 1–3, columns 1–3, total sum = 45) containing three cages fully inside Box 1 plus one 2-cell cage crossing into Box 2:

Given cages in Box 1: Cage A at r1c1 r1c2 has sum 3; Cage B at r2c1 r2c2 r3c1 has sum 24; Cage C at r2c3 r3c2 r3c3 has sum 12; and Cage D at r1c3 r1c4 has sum 15.
Step 1 — Innie calculation via Rule of 45: Cages A, B, and C cover 8 of the 9 cells of Box 1 (all except r1c3). Their combined sum is 3 + 24 + 12 = 39. Since Box 1 must sum to 45, the 9th cell is immediately solved: r1c3 = 45 − 39 = 6.
Step 2 — Outie propagation into Box 2: Cage D covers r1c3 r1c4 with target sum 15. Substituting r1c3 = 6 forces r1c4 = 15 − 6 = 9.
Step 3 — Extreme cage partitioning: Cage A (2 cells, sum 3) has only one valid non-repeating partition {1, 2}, locking digits 1 and 2 into r1c1 and r1c2. Cage B (3 cells, sum 24) has only one partition {7, 8, 9}, locking 7, 8, and 9 into r2c1, r2c2, r3c1.

Killer Sudoku Cage Size, Sum Range & Constraint Reference Table

Cage Size (Cells)	Minimum Sum (S_min)	Min Sum Digits	Maximum Sum (S_max)	Max Sum Digits	Total Distinct Digit Sets C(9, n)
1 cell	1	`[1]`	9	`[9]`	9 singletons
2 cells	3	`[1, 2]`	17	`[8, 9]`	36 pairs (4 unique sums: 3, 4, 16, 17)
3 cells	6	`[1, 2, 3]`	24	`[7, 8, 9]`	84 triples (4 unique sums: 6, 7, 23, 24)
4 cells	10	`[1, 2, 3, 4]`	30	`[6, 7, 8, 9]`	126 quads (4 unique sums: 10, 11, 29, 30)
5 cells	15	`[1, 2, 3, 4, 5]`	35	`[5, 6, 7, 8, 9]`	126 quintuplets (4 unique sums: 15, 16, 34, 35)
6 cells	21	`[1, 2, 3, 4, 5, 6]`	39	`[4, 5, 6, 7, 8, 9]`	84 sextuplets (4 unique sums: 21, 22, 38, 39)
7 cells	28	`[1, 2, 3, 4, 5, 6, 7]`	42	`[3, 4, 5, 6, 7, 8, 9]`	36 septuplets (4 unique sums: 28, 29, 41, 42)
8 cells	36	`[1..8]` (missing 9)	44	`[2..9]` (missing 1)	9 octuplets (all 9 sums 36–44 are unique)
9 cells (Full Unit)	45	`[1, 2, 3, 4, 5, 6, 7, 8, 9]`	45	`[1, 2, 3, 4, 5, 6, 7, 8, 9]`	1 complete set (Rule of 45)