10x10 Queens Puzzle #51
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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #51
What follows is a complete solve of puzzle #51, 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: last square in a region, region confined to one line, region shadow, pairing argument on rows, line confined to one region, last square in a row.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Only row 3, column 1 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 1, the rest of the blue 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 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.
Step 3. Only row 10, column 9 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 9, in sea green and in the ring around that square is now dead.
Step 4. Wherever mint ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 4 — 2 of 10 queens placed. Step 5. Pale yellow cannot leave column 2 — every one of its open squares (4 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 2 squares in all.
Step 6. The orange region has 3 squares left, spread across rows 8 and 9, columns 3 and 6. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 7. Indigo and amber are now boxed into rows 1 and 2 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 2 squares at once.
Step 8. The mint region is down to one open square, at row 4, column 3. Every colour holds exactly one queen, so that square is forced. Row 4 and column 3 are now spoken for, along with the squares touching that queen.
Step 9. Only row 9, column 6 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 6, the rest of the orange region and every square adjacent to it — diagonals included.
The board after step 9 — 4 of 10 queens placed. Step 10. Wherever slate 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 11. Red is down to 4 candidates in rows 5 and 6, columns 5, 7, and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 12. Every open square left in row 6 (2 open squares) belongs to the red region. Since that row 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: row 5 is down to one square, putting a queen on row 5, column 5; red has one square left, so its queen takes row 6, column 7; khaki has one square left, so its queen takes row 8, column 8; indigo has one square left, so its queen takes row 1, column 4; amber has one square left, so its queen takes row 2, column 10; pale yellow has one square left, so its queen takes row 7, 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 7, column 2.
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 #51
10x10 puzzle #51 is rated medium. 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 #51 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 #51 are: blue (1 square), amber (2 squares), sea green (4 squares), mint (5 squares), orange (8 squares), red (11 squares), slate (14 squares), pale yellow (14 squares), khaki (14 squares), indigo (27 squares). An even split would hand every color 10 squares, so the blue region is running 9 squares under that average while the indigo region is the roomiest area on the grid with 27 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Blue has only 1 place to put its queen; indigo has 27. The spread between them on this board is 26 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 #51, packing 100% of its 2×1 bounding box, whereas red straggles across its 5×5 box at just 44%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Here is the full footprint of every color on puzzle #51, 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 1–1. It never leaves row 3, so its queen is locked to that row.
Amber covers 2 squares, spanning 2 rows and 1 column (rows 1–2, columns 10–10). It sits wholly inside column 10, so its queen must be in that column.
Sea green occupies 4 squares within rows 8–10, columns 9–10.
Mint: 5 squares across rows 2–4, columns 3–5.
Orange covers 8 squares, spanning 3 rows and 5 columns (rows 8–10, columns 3–7).
Red occupies 11 squares within rows 4–8, columns 4–8.
Slate: 14 squares across rows 4–9, columns 2–6.
Pale yellow covers 14 squares, spanning 8 rows and 2 columns (rows 3–10, columns 1–2).
Khaki occupies 14 squares within rows 6–10, columns 6–10.
Indigo: 27 squares across rows 1–6, columns 1–10.
Where to start on this board
Open on the tightest color. Here that is blue, confined to 1 square between rows 3 and 3 and columns 1 and 1. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #51.
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. Amber is no more than two rows tall, so its queen is 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 #51
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 1, 2, 8 and 9 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 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 8 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
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.
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.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #51 is rated medium
The rating reflects how much the board concedes up front. Puzzle #51 has three 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 26 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
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 #51 the place to look for one is around row 1.
As a sanity check rather than a spoiler: the finished solution to #51 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 #51?
Puzzle #51 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #51 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #51, 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 #51?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.