10x10 Queens Puzzle #52
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 10-step reasoning is written out below.
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Step-by-step solution for puzzle #52
What follows is a complete solve of puzzle #52, worked the way a person works it rather than the way a computer does. It runs to 10 deductions, every one of them a forced placement. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the last square in a row.
Techniques this board needs: last square in a region, last square in a row.
Show the full walkthrough (10 steps) — contains the solution
Step 1. Only row 9, column 1 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 1, the rest of the orange region and every square adjacent to it — diagonals included.
Step 2. The red region is down to one open square, at row 6, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 2, the remaining red squares, and the ring of neighbours around row 6, column 2.
Step 3. Only row 1, column 3 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 3, in indigo and in the ring around that square is now dead.
The board after step 3 — 3 of 10 queens placed. Step 4. The amber region is down to one open square, at row 2, column 10. Every colour holds exactly one queen, so that square is forced. Row 2 and column 10 are now spoken for, along with the squares touching that queen.
Step 5. Only row 3, column 5 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 5, the rest of the blue region and every square adjacent to it — diagonals included.
Step 6. The mint region is down to one open square, at row 4, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 8, the remaining mint squares, and the ring of neighbours around row 4, column 8.
Step 7. Only row 10, column 4 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 4, in sea green and in the ring around that square is now dead.
The board after step 7 — 7 of 10 queens placed. Step 8. Row 5 has a single open square left, at column 6 in the slate region. Every row needs a queen, so this one is forced. Cross out the whole of row 5 and column 6, the remaining slate squares, and the ring of neighbours around row 5, column 6.
Step 9. Only row 8, column 7 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 7, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 10. The pale yellow region is down to one open square, at row 7, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 9, the remaining pale yellow squares, and the ring of neighbours around row 7, column 9.
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 #52
This is puzzle number 52 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 #52 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 #52 are: orange (1 square), indigo (3 squares), amber (3 squares), red (3 squares), blue (6 squares), mint (6 squares), sea green (13 squares), khaki (15 squares), pale yellow (17 squares), slate (33 squares). An even split would hand every color 10 squares, so the orange region is running 9 squares under that average while the slate region is the roomiest area on the grid with 33 squares.
Every color takes one queen and one only, whatever its size — which is why the 32 square difference between orange and slate is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
Look at arrangement as well as area. Orange is the densest region on this grid at 100% of a 1×1 frame; khaki is the loosest, occupying only 43% of its 5×7 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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.
Orange covers 1 square, spanning 1 row and 1 column (rows 9–9, columns 1–1). It never leaves row 9, so its queen is locked to that row.
Indigo occupies 3 squares within rows 1–2, columns 2–3.
Amber: 3 squares across rows 1–2, columns 9–10.
Red covers 3 squares, spanning 2 rows and 2 columns (rows 6–7, columns 1–2).
Blue occupies 6 squares within rows 1–3, columns 4–6.
Mint: 6 squares across rows 2–4, columns 8–10.
Sea green covers 13 squares, spanning 3 rows and 6 columns (rows 8–10, columns 1–6).
Khaki occupies 15 squares within rows 6–10, columns 2–8.
Pale yellow: 17 squares across rows 4–10, columns 8–10.
Slate covers 33 squares, spanning 7 rows and 8 columns (rows 1–7, columns 1–8).
Where to start on this board
Orange is the natural opening on #52: it is the smallest region on the grid at 1 square, boxed into rows 9–9 and columns 1–1. 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 orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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 red 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 #52
A quick way to find the next move is to count how many different colors touch each line. On this board, row 5 is the narrowest, touched by only 2 colors. Column 7 is the tightest column, 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 2, meeting 5 colors, is the opposite case — the busiest line on #52 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.
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 #52 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 #52 is rated medium
Ratings track how generous the opening is. On #52 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 32-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #52 is row 5.
Here is a shape check that gives nothing away: when #52 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 #52?
Puzzle #52 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #52 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #52, 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 #52?
Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.