10x10 Queens Puzzle #69
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 11-step reasoning is written out below.
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Step-by-step solution for puzzle #69
What follows is a complete solve of puzzle #69, worked the way a person works it rather than the way a computer does. It runs to 11 deductions — 10 queen placements and 1 elimination step 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.
Show the full walkthrough (3 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: slate has one square left, so its queen takes row 5, column 3; red has one square left, so its queen takes row 6, column 10; mint has one square left, so its queen takes row 4, column 9; pale yellow has one square left, so its queen takes row 7, column 8; indigo has one square left, so its queen takes row 1, column 7; blue has one square left, so its queen takes row 3, 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 3, column 6.
Step 2. Wherever orange ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 2 — 6 of 10 queens placed. Step 3. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: sea green has one square left, so its queen takes row 10, column 4; khaki has one square left, so its queen takes row 8, column 5; orange has one square left, so its queen takes row 9, column 1; amber has one square left, so its queen takes row 2, 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 2, column 2.
The board after step 3 — 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 #69
Puzzle #69 is one of the medium-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 #69, 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 #69 are: slate (1 square), red (2 squares), mint (4 squares), indigo (5 squares), sea green (5 squares), orange (7 squares), pale yellow (11 squares), blue (15 squares), amber (24 squares), khaki (26 squares). An even split would hand every color 10 squares, so the slate region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 26 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 slate, with 1 square, is under far more pressure than khaki with 26. A 25-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 slate is the most solid, filling 100% of its 1×1 box, and blue the most diffuse at 42% of its 6×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #69 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.
Slate: 1 square across rows 5–5, columns 3–3. It never leaves row 5, so its queen is locked to that row.
Red covers 2 squares, spanning 2 rows and 1 column (rows 5–6, columns 10–10). It sits wholly inside column 10, so its queen must be in that column.
Mint occupies 4 squares within rows 2–4, columns 9–10.
Indigo: 5 squares across rows 1–2, columns 7–10.
Sea green covers 5 squares, spanning 3 rows and 3 columns (rows 8–10, columns 2–4).
Orange occupies 7 squares within rows 6–10, columns 1–2.
Pale yellow: 11 squares across rows 4–8, columns 7–10.
Blue covers 15 squares, spanning 6 rows and 6 columns (rows 2–7, columns 4–9).
Amber occupies 24 squares within rows 1–7, columns 1–6.
Khaki: 26 squares across rows 5–10, columns 3–10.
Where to start on this board
Begin where the choices are fewest. On this board that means slate, which owns just 1 square inside a 1×1 window running from row 5 to row 5 and column 3 to column 3. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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 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 #69
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. Column 1 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 6, meeting 6 colors, is the opposite case — the busiest line on #69 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 #69 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers three single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 25. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 medium board from the rest.
Stuck on #69 despite the medium 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 #69, start hunting for it near row 1.
Here is a shape check that gives nothing away: when #69 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 #69?
Puzzle #69 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 #69 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #69, 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 #69?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.