10x10 Queens Puzzle #291
Difficulty: hard. 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 #291
What follows is a complete solve of puzzle #291, 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 pairing argument on columns.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, pairing argument on columns.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Only row 6, column 10 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 10, the rest of the red region and every square adjacent to it — diagonals included.
Step 2. The pale yellow region is down to one open square, at row 7, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 1, the remaining pale yellow squares, and the ring of neighbours around row 7, column 1.
Step 3. 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: 6 squares in all.
The board after step 3 — 2 of 10 queens placed. Step 4. All of the amber region's remaining squares (3 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
Step 5. Blue is down to 6 candidates in rows 3 and 4, columns 5–9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.
Step 6. The slate region has 5 squares left, spread across rows 3–5, columns 4 and 5. 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 7. Wherever khaki 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.
The board after step 7 — 2 of 10 queens placed. Step 8. Here is the pairing argument. The blue, mint, and slate regions have no open squares outside rows 3–5 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 4 squares come off in one stroke.
Step 9. Indigo, mint, slate, and khaki are now boxed into columns 2–5 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 8 squares at once.
Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 6; sea green has one square left, so its queen takes row 10, column 7; khaki has one square left, so its queen takes row 8, column 4; indigo has one square left, so its queen takes row 1, column 3; slate has one square left, so its queen takes row 5, column 5; orange has one square left, so its queen takes row 9, column 9; blue has one square left, so its queen takes row 3, column 8; mint has one square left, so its queen takes row 4, column 2. 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 2.
The board after step 10 — 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 #291
10x10 puzzle #291 is rated hard. Ten colored regions cover its 100 squares, and your job is to seat ten queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
There is exactly one way to finish #291, 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 #291 are: red (1 square), pale yellow (2 squares), indigo (3 squares), amber (6 squares), khaki (11 squares), slate (12 squares), sea green (12 squares), mint (13 squares), blue (17 squares), orange (23 squares). An even split would hand every color 10 squares, so the red region is running 9 squares under that average while the orange region is the roomiest area on the grid with 23 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Red is working with 1 square while orange has 23 to spare, a gap of 22. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #291, packing 100% of its 1×3 bounding box, whereas slate straggles across its 5×5 box at just 48%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Red: 1 square across rows 6–6, columns 10–10. It never leaves row 6, so its queen is locked to that row.
Pale yellow covers 2 squares, spanning 2 rows and 1 column (rows 6–7, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Indigo occupies 3 squares within rows 1–1, columns 3–5. It never leaves row 1, so its queen is locked to that row.
Amber: 6 squares across rows 1–2, columns 3–7.
Khaki covers 11 squares, spanning 4 rows and 4 columns (rows 7–10, columns 1–4).
Slate occupies 12 squares within rows 3–7, columns 2–6.
Sea green: 12 squares across rows 7–10, columns 2–7.
Mint covers 13 squares, spanning 5 rows and 3 columns (rows 1–5, columns 1–3).
Blue occupies 17 squares within rows 1–5, columns 5–10.
Orange: 23 squares across rows 4–10, columns 6–10.
Where to start on this board
Open on the tightest color. Here that is red, confined to 1 square between rows 6 and 6 and columns 10 and 10. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #291.
Because red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Indigo, amber and pale yellow 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 #291
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 3, 8, 9 and 10 are the narrowest, touched by only 3 colors. Columns 8 and 9 are the tightest columns, with 2 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 7 meets 5 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
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.
With ten queens to seat and ten regions competing over 100 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 10x10 boards quickly and players who stall halfway.
Resist the plausible-looking queen. On #291 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #291 is rated hard
Ratings track how generous the opening is. On #291 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 22-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
Stuck on #291 despite the hard 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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around column 8 on this board.
As a sanity check rather than a spoiler: the finished solution to #291 puts 4 queens on the outer border and 6 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #291?
Puzzle #291 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #291 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #291, 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 #291?
Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.