10x10 Queens Puzzle #9
Difficulty: easy. 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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #9
What follows is a complete solve of puzzle #9, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 10 queen placements and 7 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 region shadow.
Techniques this board needs: last square in a region, region shadow, region confined to one line, last square in a row.
Show the full walkthrough (9 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: blue has one square left, so its queen takes row 3, column 7; pale yellow has one square left, so its queen takes row 7, column 3; sea green has one square left, so its queen takes row 10, column 2; khaki has one square left, so its queen takes row 8, column 1. 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 8, column 1.
Step 2. Indigo is down to 5 candidates in rows 1, 2, and 4, columns 8–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. All of the orange region's remaining squares (3 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.
The board after step 3 — 4 of 10 queens placed. Step 4. Wherever mint 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 5. Red is down to 3 candidates in rows 5 and 6, columns 4–6, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.
Step 6. Slate cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 3 squares in all.
The board after step 6 — 4 of 10 queens placed. Step 7. All of the mint region's remaining squares (3 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 8. Indigo cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to indigo, so every square in it owned by another colour is dead: 1 square in all.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: orange has one square left, so its queen takes row 9, column 4; row 2 is down to one square, putting a queen on row 2, column 5; red has one square left, so its queen takes row 6, column 6; slate has one square left, so its queen takes row 5, column 8; indigo has one square left, so its queen takes row 1, column 9; mint has one square left, so its queen takes row 4, 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 4, column 10.
The board after step 9 — 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 #9
Puzzle #9 is one of the easy-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 #9, 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 #9 are: blue (1 square), pale yellow (1 square), sea green (1 square), indigo (8 squares), khaki (8 squares), slate (9 squares), mint (12 squares), red (13 squares), orange (17 squares), amber (30 squares). An even split would hand every color 10 squares, so the blue region is running 9 squares under that average while the amber region is the roomiest area on the grid with 30 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 blue, with 1 square, is under far more pressure than amber with 30. A 29-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here blue is the most solid, filling 100% of its 1×1 box, and red the most diffuse at 52% of its 5×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #9, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Blue: 1 square across rows 3–3, columns 7–7. It never leaves row 3, so its queen is locked to that row.
Pale yellow covers 1 square, spanning 1 row and 1 column (rows 7–7, columns 3–3). It never leaves row 7, so its queen is locked to that row.
Sea green occupies 1 square within rows 10–10, columns 2–2. It never leaves row 10, so its queen is locked to that row.
Indigo: 8 squares across rows 1–4, columns 8–10.
Khaki covers 8 squares, spanning 6 rows and 2 columns (rows 5–10, columns 1–2).
Slate occupies 9 squares within rows 4–7, columns 6–9.
Mint: 12 squares across rows 4–10, columns 8–10.
Red covers 13 squares, spanning 5 rows and 5 columns (rows 5–9, columns 3–7).
Orange occupies 17 squares within rows 7–10, columns 2–9.
Amber: 30 squares across rows 1–6, columns 1–9.
Where to start on this board
Begin where the choices are fewest. On this board that means blue, which owns just 1 square inside a 1×1 window running from row 3 to row 3 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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. Pale yellow 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 #9
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 2 are the narrowest, touched by only 2 colors. Columns 1 and 10 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 7, meeting 6 colors, is the opposite case — the busiest line on #9 and the last one you should expect to crack.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
The whole middle game reduces to two sweeps. Sweep the ten colors looking for one with a single square left; sweep the ten rows and ten columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 10x10 solve from a stalled one.
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 #9 is rated easy
The rating reflects how much the board concedes up front. Puzzle #9 has six regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 29 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a easy board from the rest.
Stuck on #9 despite the easy 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.
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 #9, start hunting for it near row 1.
Here is a shape check that gives nothing away: when #9 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 #9?
Puzzle #9 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #9 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #9, 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 #9?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.