10x10 Queens Puzzle #166
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 #166
What follows is a complete solve of puzzle #166, 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on columns, proof by contradiction, last square in a column.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Only row 6, column 1 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 1, the rest of the red region and every square adjacent to it — diagonals included.
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. Indigo cannot leave row 1 — every one of its open squares (3 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 5 squares in all.
Step 4. 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.
The board after step 4 — 2 of 10 queens placed. Step 5. Orange is down to 5 candidates in rows 8 and 9, columns 2, 4, and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 6. Here is the pairing argument. The indigo, slate, and pale yellow regions have no open squares outside columns 7, 8, and 10 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 7 squares come off in one stroke.
Step 7. Blue and khaki are now boxed into columns 5 and 6 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 5 squares at once.
Step 8. Orange is down to 4 candidates in rows 8 and 9, columns 2 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
The board after step 8 — 2 of 10 queens placed. Step 9. Test row 1, column 10 for indigo: place a queen there, run the forced consequences through, and the mint region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 10. 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: indigo has one square left, so its queen takes row 1, column 7; slate has one square left, so its queen takes row 5, column 8; column 5 is down to one square, putting a queen on row 8, column 5; orange has one square left, so its queen takes row 9, 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 9, column 2.
Step 11. All of the amber region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 12. 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 4; amber has one square left, so its queen takes row 2, column 3; blue has one square left, so its queen takes row 3, column 6; pale yellow has one square left, so its queen takes row 7, column 10. 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 10.
The board after step 12 — 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 #166
Board #166 of the 10x10 bank carries a hard rating. Its 100 squares are shared between ten colored regions, and the solution places ten queens: one per row, one per column, one per region, none of them side by side or corner to corner.
There is exactly one way to finish #166, 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 #166 are: red (1 square), sea green (1 square), indigo (4 squares), slate (7 squares), khaki (9 squares), amber (13 squares), blue (14 squares), orange (14 squares), pale yellow (18 squares), mint (19 squares). An even split would hand every color 10 squares, so the red region is running 9 squares under that average while the mint region is the roomiest area on the grid with 19 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Red has only 1 place to put its queen; mint has 19. The spread between them on this board is 18 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×4 box that encloses it, while mint is the most scattered at 48% of its 8×5 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.
Red covers 1 square, spanning 1 row and 1 column (rows 6–6, columns 1–1). It never leaves row 6, so its queen is locked to that row.
Sea green occupies 1 square within rows 10–10, columns 9–9. It never leaves row 10, so its queen is locked to that row.
Indigo: 4 squares across rows 1–1, columns 7–10. It never leaves row 1, so its queen is locked to that row.
Slate covers 7 squares, spanning 4 rows and 3 columns (rows 2–5, columns 7–9).
Khaki occupies 9 squares within rows 6–10, columns 5–7.
Amber: 13 squares across rows 1–3, columns 1–6.
Blue covers 14 squares, spanning 6 rows and 4 columns (rows 2–7, columns 5–8).
Orange occupies 14 squares within rows 6–10, columns 1–5.
Pale yellow: 18 squares across rows 2–10, columns 7–10.
Mint covers 19 squares, spanning 8 rows and 5 columns (rows 2–9, columns 1–5).
Where to start on this board
The most constrained color is where a solve should begin, and on #166 that is red with 1 square. It reaches from row 6 to row 6 and from column 1 to column 1 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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 sea green 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 #166
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 10 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 #166 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.
Keep alternating between two views of the board. In the region view you ask which of the ten colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 10x10 board unwinds.
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 #166 is rated hard
The hard label comes out of four measurable things about #166: five of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 18. Boards that score tight on those measures give up their first queen fast.
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.
Stuck on #166 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 #166 the place to look for one is around row 1.
Here is a shape check that gives nothing away: when #166 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 #166?
Puzzle #166 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #166 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #166, 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 #166?
Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.