10x10 Queens Puzzle #292
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 #292
What follows is a complete solve of puzzle #292, 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 line confined to one region.
Techniques this board needs: last square in a region, region confined to one line, region shadow, line confined to one region, last square in a column, last square in a row.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Only row 1, column 4 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 4, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. All of the amber region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 10 squares come off.
Step 3. The blue region has 6 squares left, spread across rows 2–4, columns 1–3. 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. Wherever mint ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 5. Red is down to 5 candidates in rows 5–8, columns 7 and 8, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.
The board after step 5 — 1 of 10 queens placed. Step 6. The orange region has 3 squares left, spread across rows 9 and 10, columns 8 and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 7. Wherever sea green ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 8. Every open square left in column 10 (8 open squares) belongs to the slate region. Since that column must contain a queen, slate is the colour that supplies it — so slate squares anywhere else on the board are out, 10 squares in all.
Step 9. Column 9 has only orange squares left (2 open squares). The column needs a queen and only orange can provide one, which pins that colour to this column and kills its squares elsewhere: 1 square.
Step 10. All of the orange region's remaining squares (2 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
The board after step 10 — 1 of 10 queens placed. Step 11. Column 8 has only red squares left (3 open squares). The column needs a queen and only red can provide one, which pins that colour to this column and kills its squares elsewhere: 2 squares.
Step 12. Every open square left in column 7 (3 open squares) belongs to the khaki region. Since that column must contain a queen, khaki is the colour that supplies it — so khaki squares anywhere else on the board are out, 7 squares in all.
Step 13. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: column 5 is down to one square, putting a queen on row 4, column 5; amber has one square left, so its queen takes row 2, column 6; blue has one square left, so its queen takes row 3, column 1; row 5 is down to one square, putting a queen on row 5, 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 5, column 10.
Step 14. Sea green cannot leave row 10 — every one of its open squares (2 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 3 squares in all.
Step 15. 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: orange has one square left, so its queen takes row 9, column 9; khaki has one square left, so its queen takes row 8, column 7; red has one square left, so its queen takes row 6, column 8; pale yellow has one square left, so its queen takes row 7, column 2; sea green has one square left, so its queen takes row 10, column 3. 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 10, column 3.
The board after step 15 — 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 #292
This is puzzle number 292 in the 10x10 collection, rated medium. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Puzzle #292 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #292 are: indigo (1 square), orange (3 squares), amber (6 squares), sea green (6 squares), red (8 squares), blue (11 squares), mint (11 squares), pale yellow (11 squares), khaki (17 squares), slate (26 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the slate region is the roomiest area on the grid with 26 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So indigo, with 1 square, is under far more pressure than slate with 26. A 25-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; khaki is the loosest, occupying only 47% of its 6×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a medium board.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 4–4). It never leaves row 1, so its queen is locked to that row.
Orange occupies 3 squares within rows 9–10, columns 8–9.
Amber: 6 squares across rows 1–3, columns 5–7.
Sea green covers 6 squares, spanning 2 rows and 4 columns (rows 9–10, columns 1–4).
Red occupies 8 squares within rows 5–8, columns 6–8.
Blue: 11 squares across rows 1–4, columns 1–4.
Mint covers 11 squares, spanning 3 rows and 5 columns (rows 3–5, columns 2–6).
Pale yellow occupies 11 squares within rows 5–9, columns 1–3.
Khaki: 17 squares across rows 5–10, columns 3–8.
Slate covers 26 squares, spanning 10 rows and 4 columns (rows 1–10, columns 7–10).
Where to start on this board
Indigo is the natural opening on #292: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 4–4. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. Then look at which colors just lost ground: on a grid this size that one observation usually forces a second region within a move or two.
Work outward to the other flat regions next. Sea green and orange 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 #292
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2 and 4 are the narrowest, touched by only 3 colors. Column 10 is the tightest column, with 1 color.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 9 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
Two checks, run alternately, will carry you through the 100 squares of this grid. First: has any of the ten regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 10x10 boards almost always run one of those checks and forget the other.
A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.
Why puzzle #292 is rated medium
The medium label comes out of four measurable things about #292: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 25. Boards that score tight on those measures give up their first queen fast.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #292, start hunting for it near column 10.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 6 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #292?
Puzzle #292 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #292 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #292, 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 #292?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 4–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.