Summary: After the N×N rules, the fills that place a symbol without guessing are last-symbol in a line, Naked Singles, and Hidden Singles. A Naked Single is a cell whose row and column leave one leftover. A Hidden Single is a symbol that has only one legal cell left in a row or a column. There are no boxes. Each section below works one 5×5.
Start from the rules, then scan
A Latin Square is an N×N grid that holds each of N symbols once in every row and once in every column. Sudoku adds boxes on top of that rule. Word Squares is a letter Latin square with one highlighted word. This page is the grid fill only.
If a row or column still has two or more empties, “fill the last hole” is not enough. You need a scan that either asks what can this cell be? or where can this symbol go?
Those two questions are the whole page. Houses here are rows and columns only. Do not hunt a 2×2, a 3×3 block, or any other region. For the rules and the first 3×3 walkthrough, see How Latin Squares Work.
Count rows down from the top and columns left to right. A printed symbol stays put. Blank cells are empty. The worked grids are 5×5, so the symbols are 1 through 5. Each figure shows the same grid twice: on the left the scan is set up, with the cell it resolves tinted amber; on the right that one symbol is written in. Every other cell is left blank on purpose.
Last symbol in a row or column
Row/column elimination starts with a line that is already missing only one symbol. List what that row or column already shows. The leftover is the fill.
You find it by staring at one row or one column. If four of the five symbols are already there, write the fifth in the last empty. You do not need to look at the other house yet.
Worked example
Look at column 1. The amber cell is row 5, column 1 — the last empty in that column.
Column 1 already shows 1, 2, 3, and 5. The leftover is 4. Write it.
Row 5 is still empty except for that cell, so this is not “the last hole in the row.” It is the last hole in the column. Same scan, other house.
A 5×5 Level 1 page is mostly this fill. When every line still has two or more empties, switch to the two scans below.
Naked Single: one cell, one leftover
A Naked Single is an empty cell whose row and column together already show four of the five symbols. The leftover is the only one that can go in that cell.
You find it by staring at one cell. List every symbol that already appears in its row or its column. If four symbols are taken, write the fifth. You do not need to ask where else that symbol could go.
This is the same leftover test as last-symbol, used when the cell is not the last empty in its row or column. The row knocks some symbols out. The column knocks the rest out. There is no box to check.
Worked example
Look at the amber cell in row 2, column 2.
Symbols that already see that cell:
- Row 2: 4 at column 1
- Column 2: 5 at row 1, 3 at row 3, 2 at row 4
That is 2, 3, 4, and 5. The amber cell cannot take any of them. Write 1.
Row 2 still has three other empties. Column 2 still has the bottom cell empty. This scan never checked those. It only proved that this cell has no other symbol left. That is why the single is called naked: the leftover is sitting on the cell itself.
The same 1 might still be legal in another empty of row 2. That does not matter for this placement.
Want a 5×5 to try Hidden Singles on? The free demo prints one page. Puzzle Maker Pro Latin Squares is the generator; the create walkthrough is How to Get Started with Puzzle Maker Pro – Latin Squares.
Hidden Single: uniqueness — one symbol, one home
A Hidden Single is a symbol that can go in only one empty cell of a single row or a single column, even when that cell still looks like it could take several symbols.
You find it by staring at one symbol in one line. Pick a row or a column. Pick a missing symbol. Cross off every empty that already sees that symbol in its other house (the crossing column, or the crossing row). If one empty remains, write the symbol there.
The cell is not a Naked Single. It still has other leftovers. The symbol is hidden among them until you ask where that symbol is allowed to live. That is uniqueness: each symbol once per row and once per column, used as a placement, not only as a check at the end.
Worked example
Look at row 3. Ask where 4 can go in that row. The amber cell is row 3, column 1.
Row 3 is all empty. Its five cells are the amber cell, then columns 2, 3, 4, and 5.
- Column 2 already has a 4 at row 2, so that empty cannot be 4.
- Column 3 already has a 4 at row 4, so that empty cannot be 4.
- Column 4 already has a 4 at row 1, so that empty cannot be 4.
- Column 5 already has a 4 at row 5, so that empty cannot be 4.
The amber cell is the only empty in row 3 that does not already see a 4. Write 4 there.
That cell is not a Naked Single. Its row is empty and its column is empty, so 1, 2, 3, 4, and 5 all look possible if you only ask “what can this cell be?” The 4 appears when you ask “where can 4 go in this row?”
The same 4 is also the only legal home in column 1 — every other row in that column already sees a 4. Uniqueness fired in both houses. Still one placement.
Which question are you asking?
- Last symbol — Which row or column is missing only one symbol? Write it in the last empty of that line.
- Naked Single — What symbols are still legal in this cell? You have a placement when its row and column together already show the other four.
- Hidden Single — Where can this symbol go in this row or column? You have a placement when every other empty in that line already sees the symbol.
A line with one empty cell is last-symbol and Naked Single at once. How Latin Squares Work starts there. Use this page once two or more empties remain.
Scan last-symbol first, then Naked Singles — they look at one cell. When a cell still has several leftovers, switch to Hidden Singles and walk the missing symbols line by line.
Do not guess. If two cells in a line still accept the same symbol, that symbol is not a Hidden Single yet. Leave it and look at another line.
Do not invent a box scan. Pointing pairs and box-line reduction need a region. A Latin Square has none.
When two cells in a row or column share the same two leftovers, that is a pair — it does not write a symbol. On an 8×8 or 9×9, two rows that hold a leftover in the same two columns can lock that leftover the same way. Both are real row-and-column locks. This page stays on fills.
Practice on a fresh 5×5
Print a 5×5 Level 1 or Level 2 page, cover the solution, and place every last-symbol and Naked Single you can see. Those two levels finish with those two scans — they are the practice set for the first half of this page.
For Hidden Singles, print a 5×5 Level 3. Level 1 and Level 2 never leave a leftover that only Hidden Single can place, so a reader who stays on those banks has nothing left to practice the third scan.
Puzzle Maker Pro Latin Squares prints 3×3 through 9×9 grids with a matching answer key. The generate walkthrough is How to Get Started with Puzzle Maker Pro – Latin Squares. The free demo generates one page at a time if you want to try a board first.
FAQ
What is row/column elimination on a Latin Square?
Crossing off every symbol that already appears in a cell’s row or column. When one leftover remains, that cell is a Naked Single. When a line has one empty, the leftover for that line is the last-symbol fill.
What is a Naked Single on a Latin Square?
An empty cell whose row and column already contain four different symbols. The fifth symbol is the only one that can go in that cell. There is no box to include in the count.
What is a Hidden Single on a Latin Square?
A symbol that has only one legal empty cell left in a single row or a single column. That cell may still look like it could hold other symbols until you ask where the chosen symbol is allowed to go.
Do I use boxes?
No. A Latin Square has no boxes and no regions. Sudoku keeps the same row-and-column rule and then adds boxes. If you are hunting a block, you are on a Sudoku page.
Is this the same as Word Squares?
No. Word Squares is a letter Latin square with one highlighted word on a row or a column. This page is the Latin Squares fill only.
Do I need pencil marks?
No. Pencil marks make Hidden Singles faster because you can see a symbol that appears in only one cell of a line. You can also find the same placement by walking a missing symbol across the empties of that line, as in the row-3 example.