BTT Mathematics / Primary Mathematics Learning Hub / Latin Squares and Constraint Propagation
A Latin square of order n is an n×n grid filled with n symbols so that every symbol appears exactly once in each row and exactly once in each column. A standard Sudoku adds extra region constraints on top of the row-and-column structure.
For example, the cyclic order-3 Latin square
1 2 3
2 3 1
3 1 2
contains each symbol once in every row and column. The rule is global, but many deductions are local: if a row already contains 1 and2, its remaining cell must be3.
This guide is Primary Mathematics enrichment. It develops constraint reasoning, candidate sets, contradiction checks and systematic search. It connects to Simultaneous Conditions and Multiple Constraints, Logic, Deduction and Truth Conditions and Working Systematically.
Standard 9×9 Sudoku is not treated as a syllabus requirement. Practice uses small Latin squares and 4×4 Sudoku-style grids so the logic remains visible.
Latin-square structure · Candidates · Mini-Sudoku regions · Constraint propagation · Search and contradiction · 24 questions · Worked answers
1. Rows and columns are simultaneous all-different constraints
In an order-4 Latin square using symbols 1,2,3,4, each row must contain all four symbols once, and so must each column.
Worked example A: One missing value
Row 2 is 2,3,4,_. The only missing symbol is 1.
Worked example B: Column check
A candidate value can fit a row but still fail its column. Every placement must satisfy both constraints.
Worked example C: Cyclic construction
Starting from row 1,2,3,4 and shifting left by one each row gives:
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3
This is a valid order-4 Latin square.
Worked example D: Duplicate failure
Row 1,2,2,4 is invalid because symbol2 repeats and symbol3 is missing. “Four filled cells” is not enough; the all-different condition matters.
2. Candidate sets record what remains possible
Suppose a 4×4 Latin grid uses 1–4. If a blank cell lies in a row already containing1 and3, and its column already contains4, then the only remaining candidate is 2.
Worked example E: Intersection of restrictions
Row restriction allows {2,4}. Column restriction allows {1,2}. Their intersection is {2}, so the cell is forced to2.
Worked example F: Two candidates remain
If a cell can be2 or4, do not guess merely because one looks plausible. Keep both until another constraint removes one.
Worked example G: Empty candidate set
If row and column restrictions eliminate all four symbols, the current partial grid is inconsistent. At least one earlier placement is wrong.
Worked example H: Hidden single in a row
If three blank cells in a row have candidates {1,2}, {1,2,3}, {1,2}, then symbol3 can appear only in the middle cell. That cell is therefore forced to3 even though it had several candidates locally.
3. Mini-Sudoku adds region constraints
In a 4×4 Sudoku-style grid using1–4, each row, each column and each 2×2 box must contain1–4 exactly once.
Worked example I: Box single
A 2×2 box contains1,2,3 and one blank. The blank must be 4.
Worked example J: Row and box together
A cell’s row permits {2,4}, while its box already contains4. The cell must be 2.
Worked example K: Valid Latin square but invalid Sudoku
A grid can satisfy every row and column yet fail a 2×2 box. Sudoku structure is stricter than Latin-square structure.
Worked example L: Completed 4×4 Sudoku
1 2 3 4
3 4 1 2
2 1 4 3
4 3 2 1
Every row, column and 2×2 box contains1,2,3,4.
4. Constraint propagation means one forced value creates new information
When a forced value is placed, remove that value from candidate sets in the same row, column and, for Sudoku, box. This may create another forced value, which propagates the deduction.
Worked example M: A short chain
Cell A has candidates {2}; place2. Cell B in the same row had {2,4}, so it becomes {4}; place4. Cell C in B’s column had {1,4}, so it becomes {1}. One certainty generated a chain of three deductions.
Worked example N: Propagation stops safely
If every blank still has two or more candidates and no row/column/box has a unique location for a missing symbol, propagation has reached a stable state. That does not mean the puzzle is impossible; it means a different deduction or bounded search is needed.
Worked example O: Preserve candidate accuracy
Removing a candidate without a valid row, column or region reason can create false certainty. Candidate notes are only as reliable as the constraints used to update them.
5. Search should branch on a small uncertainty and verify consequences
If pure propagation stops, choose a cell with a small candidate set, temporarily assume one candidate, propagate, and check for contradiction.
Worked example P: Contradiction branch
A cell has {2,4}. Assume2. Later a row is forced to contain two 3s or a cell loses every candidate. The assumption is impossible, so the original cell must be4.
Worked example Q: Search tree
Record each assumption explicitly. If both branches lead to valid completions, the puzzle has multiple solutions. If exactly one survives, the solution is unique under the stated constraints.
Worked example R: Do not confuse search with random guessing
A controlled branch is reversible, documented and checked. Random guessing forgets why a choice was made and makes error repair difficult.
Worked example S: Existence versus uniqueness
Finding one completed Latin square proves existence. It does not prove uniqueness unless all alternative branches are eliminated.
6. Practice: 24 original questions
Questions 1–8: Latin-square structure
1. In an order-3 Latin square, how many symbols are used?
2. Row 2,3,_ uses symbols1–3. Find the missing value.
3. Is row1,2,2,4 valid in an order-4 Latin square?
4. Complete the cyclic row after1,2,3,4 by shifting left once.
5. Complete the next cyclic row after2,3,4,1.
6. A blank’s row allows {2,4}; its column allows {1,2}. Find the value.
7. What does an empty candidate set tell you?
8. In one row, candidates are {1,2},{1,2,3},{1,2}. Where must3 go?
Questions 9–16: Mini-Sudoku constraints
9. A 2×2 box contains1,2,3,_. Find the blank.
10. A cell has row candidates {2,4}, but its box already contains4. Find the cell.
11. Does every completed 4×4 Latin square automatically satisfy 2×2 Sudoku boxes?
12. How many times must each symbol appear in each row of a 4×4 Sudoku?
13. How many times in each column?
14. How many times in each 2×2 box?
15. Check whether row3,4,1,2 contains all four symbols exactly once.
16. In the completed grid shown in the guide, what is the entry at row3,column2?
Questions 17–24: Propagation and search
17. A cell has candidates {2}. What should propagation do first?
18. In the same row, another cell had {2,4}. After placing2, what remains?
19. If a trial assumption eventually gives a cell no candidates, what can you conclude about the assumption?
20. If both branches of a two-candidate choice lead to valid completions, what does that imply?
21. If exactly one branch survives, what does that show about that choice?
22. Why is a documented branch different from random guessing?
23. Does finding one valid completion prove uniqueness?
24. State the three constraint types used in standard 4×4 mini-Sudoku.
7. Worked answers
Answers 1–8
1. 3 symbols.
2. 1.
3. No. Symbol2 repeats and3 is missing.
4. 2,3,4,1.
5. 3,4,1,2.
6. 2. It is the intersection of the allowed sets.
7. The partial assignment is inconsistent. At least one earlier assumption or placement must be revised.
8. The middle cell. It is the only candidate set containing3.
Answers 9–16
9. 4.
10. 2.
11. No. Sudoku adds region constraints beyond the Latin-square row/column conditions.
12. Once.
13. Once.
14. Once.
15. Yes.
16. 1.
Answers 17–24
17. Place2.
18. {4}. The2 candidate is removed.
19. The assumption is impossible.
20. More than one solution is compatible with the current puzzle constraints.
21. The surviving candidate is forced.
22. It is recorded, reversible and checked by propagated consequences.
23. No. Other branches may contain different valid completions.
24. Row, column and 2×2 box constraints.
8. Teaching and transfer
If a learner fills a cell because it “looks right,” ask for the candidate set and the specific constraint that removes each alternative.
When row reasoning ignores columns
Write the allowed set from the row and from the column separately, then intersect them. This makes simultaneous constraints visible.
When Sudoku rules blur together
Use three labels beside each cell: row, column, box. A candidate survives only if all three permit it.
When propagation stalls
Do not force a value. Search for hidden singles or choose a cell with the smallest candidate set for a documented branch.
When a contradiction appears
Return to the most recent unsupported assumption rather than changing a forced deduction earlier in the chain.
Connect to algorithms
Candidate elimination, propagation and branching form a finite decision procedure, linking directly to Mathematical Algorithms, Flowcharts and Decision Trees.
Continue through this enrichment collection
For one-sided surfaces, use Möbius Strips, One-Sided Surfaces and Topology. For codeword comparison, use Hamming Distance, Error-Correcting Codes and Parity Checks. For geometric enclosure, use Convex Hulls, Extreme Points and Geometric Enclosure.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Mini-Sudoku examples use 4×4 grids to expose constraint logic clearly.