Sudoku as a Constraint Satisfaction Problem

Sudoku is a compact CSP: every empty cell is a variable, each domain is a subset of digits 1–9, and row, column, and box constraints require distinct values.

Model

Variables: one for each empty cell, named by row and column.
Domains: initially {1,…,9}, reduced by clues.
Constraints: cells in each row, column, and 3×3 box must be different.

r1c15	r1c23	r1c3	r1c4	r1c57	r1c6	r1c7	r1c8	r1c9
r2c16	r2c2	r2c3	r2c41	r2c59	r2c65	r2c7	r2c8	r2c9
r3c1	r3c29	r3c38	r3c4	r3c5	r3c6	r3c7	r3c86	r3c9
r4c18	r4c2	r4c3	r4c4	r4c56	r4c6	r4c7	r4c8	r4c93
r5c14	r5c2	r5c3	r5c48	r5c5	r5c63	r5c7	r5c8	r5c91
r6c17	r6c2	r6c3	r6c4	r6c52	r6c6	r6c7	r6c8	r6c96
r7c1	r7c26	r7c3	r7c4	r7c5	r7c6	r7c72	r7c88	r7c9
r8c1	r8c2	r8c3	r8c44	r8c51	r8c69	r8c7	r8c8	r8c95
r9c1	r9c2	r9c3	r9c4	r9c58	r9c6	r9c7	r9c87	r9c99

The running puzzle

Every page uses this puzzle. Blue cells are the variables currently discussed. For example, fixed digits in row 1, column 3, and the upper-left box reduce r1c3 to {1,2,4}. The clues reduce r5c5 to the singleton {5}.

Binary and global formulations

A binary model adds Xi ≠ Xj for every pair of peers. A global model uses 27 AllDifferent constraints: nine rows, nine columns, and nine boxes.

Check: How many peers does a center cell have? Do not count a cell twice when it shares both a row and a box.