10x10 Queens Puzzle #17
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 #17
What follows is a complete solve of puzzle #17, 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, line confined to one region, last square in a column, pairing argument on rows.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Only row 1, column 7 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 7, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The orange region is down to one open square, at row 9, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 6, the remaining orange squares, and the ring of neighbours around row 9, column 6.
Step 3. Amber cannot leave column 1 — 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: 8 squares in all.
Step 4. Wherever blue 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.
Step 5. Mint is down to 3 candidates in rows 2–4, columns 2 and 3, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about mint itself.
The board after step 5 — 2 of 10 queens placed. Step 6. Every open square left in column 10 (8 open squares) belongs to the pale yellow region. Since that column must contain a queen, pale yellow is the colour that supplies it — so pale yellow squares anywhere else on the board are out, 9 squares in all.
Step 7. Scan column 9 and only row 10, column 9 remains — a sea green square. Every column takes a queen, so it takes this one. 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 8. Khaki is down to 4 candidates in rows 6–8, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 9. Column 8 has only slate squares left (3 open squares). The column needs a queen and only slate can provide one, which pins that colour to this column and kills its squares elsewhere: 2 squares.
Step 10. Here is the pairing argument. The amber and blue regions have no open squares outside rows 2 and 3 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 5 squares come off in one stroke.
The board after step 10 — 3 of 10 queens placed. Step 11. 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: mint has one square left, so its queen takes row 4, column 2; amber has one square left, so its queen takes row 2, column 1; blue has one square left, so its queen takes row 3, column 5; slate has one square left, so its queen takes row 5, column 8. 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 5, column 8.
Step 12. Red cannot leave column 3 — every one of its open squares (3 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 1 square in all.
Step 13. The khaki region is down to one open square, at row 8, column 4. Every colour holds exactly one queen, so that square is forced. Row 8 and column 4 are now spoken for, along with the squares touching that queen.
Step 14. Only row 6, column 3 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 3, the rest of the red region and every square adjacent to it — diagonals included.
Step 15. The pale yellow region is down to one open square, at row 7, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 10, the remaining pale yellow squares, and the ring of neighbours around row 7, column 10.
The board after step 15 — 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 #17
Puzzle #17 is one of the hard-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 #17, 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 #17 are: indigo (1 square), blue (4 squares), orange (4 squares), amber (5 squares), sea green (5 squares), mint (7 squares), red (15 squares), slate (18 squares), khaki (18 squares), pale yellow (23 squares). An even split would hand every color 10 squares, so the indigo region is running 9 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. Indigo has only 1 place to put its queen; pale yellow has 23. The spread between them on this board is 22 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 46% of its 10×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #17 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.
Indigo occupies 1 square within rows 1–1, columns 7–7. It never leaves row 1, so its queen is locked to that row.
Blue: 4 squares across rows 1–3, columns 4–5.
Orange covers 4 squares, spanning 3 rows and 2 columns (rows 8–10, columns 6–7).
Amber occupies 5 squares within rows 1–3, columns 1–3.
Sea green: 5 squares across rows 8–10, columns 8–9.
Mint covers 7 squares, spanning 4 rows and 3 columns (rows 2–5, columns 1–3).
Red occupies 15 squares within rows 3–8, columns 1–4.
Slate: 18 squares across rows 1–6, columns 5–8.
Khaki covers 18 squares, spanning 5 rows and 6 columns (rows 6–10, columns 1–6).
Pale yellow occupies 23 squares within rows 1–10, columns 6–10.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 7 to column 7. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: blue, orange and sea green span 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 #17
A quick way to find the next move is to count how many different colors touch each line. On this board, row 7 is the narrowest, touched by only 3 colors. Column 10 is the tightest column, with 1 color.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 3 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
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.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #17 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 22. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
Stuck on #17 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.
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 #17 the place to look for one is around column 10.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 6 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #17?
Puzzle #17 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #17 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #17, 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 #17?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.