10x10 Queens Puzzle #251
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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #251
What follows is a complete solve of puzzle #251, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 10 queen placements and 8 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 rows.
Techniques this board needs: region confined to one line, last square in a region, pairing argument on rows, last square in a row, region shadow, last square in a column, line confined to one region.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Amber cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 10 squares in all.
Step 2. The sea green region is down to one open square, at row 10, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 9, the remaining sea green squares, and the ring of neighbours around row 10, column 9.
Step 3. Slate cannot leave column 8 — every one of its open squares (3 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 5 squares in all.
Step 4. Here is the pairing argument. The indigo, amber, and blue regions have no open squares outside rows 1–3 — 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: 6 squares come off in one stroke.
The board after step 4 — 1 of 10 queens placed. Step 5. Only row 5, column 8 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 8, the rest of the slate region and every square adjacent to it — diagonals included.
Step 6. All of the mint region's remaining squares (2 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
Step 7. Look along row 3: one square survives, row 3, column 1, which belongs to blue. A row cannot be left empty, so the queen goes there. That clears row 3, column 1, the rest of the blue region and every square adjacent to it — diagonals included.
Step 8. Red is down to 4 candidates in rows 6 and 7, columns 2 and 4, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.
Step 9. The orange region has 3 squares left, spread across rows 8 and 9, columns 2 and 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.
The board after step 9 — 3 of 10 queens placed. Step 10. Column 3 now has exactly one open square, row 9, column 3, in the orange region. Columns must each hold a queen, so that placement is forced. That clears row 9, column 3, the rest of the orange region and every square adjacent to it — diagonals included.
Step 11. Khaki cannot leave row 8 — every one of its open squares (2 open squares) sits there. That row belongs to khaki, so every square on it owned by another colour is dead: 3 squares in all.
Step 12. Every open square left in column 2 (2 open squares) belongs to the red region. Since that column must contain a queen, red is the colour that supplies it — so red squares anywhere else on the board are out, 2 squares in all.
Step 13. 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: column 4 is down to one square, putting a queen on row 1, column 4; amber has one square left, so its queen takes row 2, column 10; column 7 is down to one square, putting a queen on row 7, column 7; red has one square left, so its queen takes row 6, column 2; khaki has one square left, so its queen takes row 8, column 5; mint has one square left, so its queen takes row 4, column 6. 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 6.
The board after step 13 — 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 #251
10x10 puzzle #251 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.
Puzzle #251 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 #251 are: amber (2 squares), sea green (5 squares), mint (6 squares), orange (8 squares), blue (9 squares), khaki (9 squares), indigo (10 squares), slate (11 squares), red (17 squares), pale yellow (23 squares). An even split would hand every color 10 squares, so the amber region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 23 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 2 places to put its queen; pale yellow has 23. The spread between them on this board is 21 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Square count is only half the story — how a color is arranged counts too. Amber is the tidiest shape on #251, packing 100% of its 2×1 bounding box, whereas khaki straggles across its 5×4 box at just 45%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #251 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Amber occupies 2 squares within rows 1–2, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.
Sea green: 5 squares across rows 7–10, columns 9–10.
Mint covers 6 squares, spanning 3 rows and 4 columns (rows 3–5, columns 5–8).
Orange occupies 8 squares within rows 8–10, columns 1–3.
Blue: 9 squares across rows 1–3, columns 1–5.
Khaki covers 9 squares, spanning 5 rows and 4 columns (rows 6–10, columns 3–6).
Indigo occupies 10 squares within rows 1–3, columns 4–9.
Slate: 11 squares across rows 2–6, columns 8–10.
Red covers 17 squares, spanning 5 rows and 4 columns (rows 3–7, columns 1–4).
Pale yellow occupies 23 squares within rows 5–10, columns 5–9.
Where to start on this board
Open on the tightest color. Here that is amber, confined to 2 squares between rows 1 and 2 and columns 10 and 10. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #251.
Because amber stays inside column 10, its queen belongs to that column and no other region may use it. Cross out the rest of column 10 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: sea green spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #251
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 4 are the narrowest, touched by only 3 colors. Columns 1, 2, 7 and 10 are the tightest columns, with 3 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 3 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.
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.
Resist the plausible-looking queen. On #251 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 #251 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 21. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #251 the place to look for one is around row 1.
As a sanity check rather than a spoiler: the finished solution to #251 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 #251?
Puzzle #251 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #251 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #251, 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 #251?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.