10x10 Queens Puzzle #81
Difficulty: medium. Place 10 queens on this 10x10 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 22-step reasoning is written out below.
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Step-by-step solution for puzzle #81
What follows is a complete solve of puzzle #81, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 10 queen placements and 12 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the pairing argument on columns.
Techniques this board needs: region shadow, region confined to one line, last square in a region, pairing argument on columns, last square in a column, line confined to one region, last square in a row.
Show the full walkthrough (18 steps) — contains the solution
Step 1. Wherever blue ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Mint is down to 5 candidates in rows 2–5, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about mint itself.
Step 3. The pale yellow region has 5 squares left, spread across rows 5–8, columns 9 and 10. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 4. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.
Step 5. Indigo cannot leave row 1 — every one of its open squares (3 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 7 squares in all.
The board after step 5 — 0 of 10 queens placed. Step 6. The amber region is down to one open square, at row 2, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 3, the remaining amber squares, and the ring of neighbours around row 2, column 3.
Step 7. Mint cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 6 squares in all.
Step 8. The red region is down to one open square, at row 6, column 2. Every colour holds exactly one queen, so that square is forced. Row 6 and column 2 are now spoken for, along with the squares touching that queen.
Step 9. Only row 4, column 1 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 1, the rest of the mint region and every square adjacent to it — diagonals included.
The board after step 9 — 3 of 10 queens placed. Step 10. Wherever sea green ends up among its 5 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 11. Orange is down to 4 candidates in rows 7–9, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 12. Here is the pairing argument. The blue and pale yellow regions have no open squares outside columns 9 and 10 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 8 squares come off in one stroke.
Step 13. Only row 1, column 8 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 8, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 14. Khaki is down to 3 candidates in rows 7 and 8, columns 6 and 7, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about khaki itself.
The board after step 14 — 4 of 10 queens placed. Step 15. Slate cannot leave row 5 — every one of its open squares (4 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 2 squares in all.
Step 16. Column 9 now has exactly one open square, row 3, column 9, in the blue region. Columns must each hold a queen, so that placement is forced. Everything else in row 3, in column 9, in blue and in the ring around that square is now dead.
Step 17. Row 10 has only sea green squares left (3 open squares). The row needs a queen and only sea green can provide one, which pins that colour to this row and kills its squares elsewhere: 1 square.
Step 18. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: row 9 is down to one square, putting a queen on row 9, column 4; sea green has one square left, so its queen takes row 10, column 6; khaki has one square left, so its queen takes row 8, column 7; slate has one square left, so its queen takes row 5, column 5; pale yellow has one square left, so its queen takes row 7, column 10. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 7, column 10.
The board after step 18 — 10 of 10 queens placed.
That completes the board: 10 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.
About 10x10 Queens puzzle #81
Puzzle #81 is one of the medium-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
There is exactly one way to finish #81, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these ten regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #81 are: blue (3 squares), indigo (5 squares), mint (5 squares), pale yellow (5 squares), sea green (6 squares), red (7 squares), amber (10 squares), orange (14 squares), khaki (15 squares), slate (30 squares). An even split would hand every color 10 squares, so the blue region is running 7 squares under that average while the slate region is the roomiest area on the grid with 30 squares.
Every color takes one queen and one only, whatever its size — which is why the 27 square difference between blue and slate is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 83% of its 2×3 box, and slate the most diffuse at 54% of its 7×8 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Blue: 3 squares across rows 3–4, columns 9–10.
Indigo covers 5 squares, spanning 2 rows and 3 columns (rows 1–2, columns 8–10).
Mint occupies 5 squares within rows 2–5, columns 1–2.
Pale yellow: 5 squares across rows 5–8, columns 9–10.
Sea green covers 6 squares, spanning 2 rows and 5 columns (rows 9–10, columns 3–7).
Red occupies 7 squares within rows 5–8, columns 1–3.
Amber: 10 squares across rows 1–3, columns 1–6.
Orange covers 14 squares, spanning 4 rows and 5 columns (rows 7–10, columns 1–5).
Khaki occupies 15 squares within rows 6–10, columns 6–10.
Slate: 30 squares across rows 1–7, columns 2–9.
Where to start on this board
Begin where the choices are fewest. On this board that means blue, which owns just 3 squares inside a 2×2 window running from row 3 to row 4 and column 9 to column 10. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Blue does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.
Work outward to the other flat regions next. Indigo and sea green are no more than two rows tall, so their queens are confined to a narrow band. When two flat regions overlap the same pair of rows, those rows are spoken for and every other color is locked out of both — the single most productive pattern in the game, and it is live on this board.
The tightest lines on puzzle #81
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4, 9 and 10 are the narrowest, touched by only 3 colors. Columns 7 and 8 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 7, meeting 5 colors, is the opposite case — the busiest line on #81 and the last one you should expect to crack.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
Keep alternating between two views of the board. In the region view you ask which of the ten colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 10x10 board unwinds.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #81 is rated medium
Ratings track how generous the opening is. On #81 you get no region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 3 colors, and a 27-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
Stuck on #81 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
Techniques that work well at 10x10
At 10x10 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #81 is row 1.
Here is a shape check that gives nothing away: when #81 is solved, 4 queens sit on the outer edge of the grid and 6 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
How to play Queens
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
- There is always exactly one valid solution. Use logic, never guessing, to find it.
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Questions about this puzzle
How hard is 10x10 Queens puzzle #81?
Puzzle #81 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 10x10 puzzle #81 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #81, has been verified to have exactly one valid arrangement of ten queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #81?
Start with the blue region — the most constrained color on this board, with 3 squares spanning rows 3–4 and columns 9–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.