11x11 Queens Puzzle #238
Difficulty: hard. Place 11 queens on this 11x11 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 31-step reasoning is written out below.
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Step-by-step solution for puzzle #238
What follows is a complete solve of puzzle #238, worked the way a person works it rather than the way a computer does. It runs to 31 deductions — 11 queen placements and 20 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 shadow, region confined to one line, pairing argument on rows, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (27 steps) — contains the solution
Step 1. Only row 7, column 6 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 6, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 2. The teal region is down to one open square, at row 11, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 10, the remaining teal squares, and the ring of neighbours around row 11, column 10.
Step 3. The amber region has 3 squares left, spread across rows 1 and 2, columns 9 and 11. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 4. All of the mint region's remaining squares (3 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 5. Amber cannot leave column 9 — 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: 7 squares in all.
Step 6. The khaki region has 6 squares left, spread across rows 4, 5, 6, 8, 9, and 10, columns 1 and 2. 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. Wherever orange ends up among its 7 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 7 — 2 of 11 queens placed. Step 8. Here is the pairing argument. The indigo and amber regions have no open squares outside rows 1 and 2 — 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: 10 squares come off in one stroke.
Step 9. Indigo, slate, and sea green are now boxed into columns 5, 7, and 8 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 6 squares at once.
Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 7. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 11. Test row 2, column 7 for indigo: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 12. The indigo region is down to one open square, at row 1, column 5. Every colour holds exactly one queen, so that square is forced. Row 1 and column 5 are now spoken for, along with the squares touching that queen.
Step 13. Only row 2, column 9 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 9, the rest of the amber region and every square adjacent to it — diagonals included.
Step 14. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 1. Follow the forced moves that would create and row 8 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 14 — 4 of 11 queens placed. Step 15. Test row 3, column 3 for blue: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 16. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 4. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 17. Test row 4, column 1 for blue: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 18. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 3. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Test row 4, column 4 for blue: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 20. Blue is down to 4 candidates in rows 3, 5, and 6, columns 1–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
The board after step 20 — 4 of 11 queens placed. Step 21. Test row 5, column 3 for blue: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 22. Now a short proof by contradiction. Suppose the blue queen went to row 5, column 4. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 23. Look along row 5: one square survives, row 5, column 8, which belongs to slate. A row cannot be left empty, so the queen goes there. That clears row 5, column 8, the rest of the slate region and every square adjacent to it — diagonals included.
Step 24. The sea green region is down to one open square, at row 10, column 7. Every colour holds exactly one queen, so that square is forced. Row 10 and column 7 are now spoken for, along with the squares touching that queen.
Step 25. Wherever blue ends up among its 2 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 26. Khaki is down to 3 candidates in rows 4, 8, and 9, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 27. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: orange has one square left, so its queen takes row 9, column 4; red has one square left, so its queen takes row 6, column 3; blue has one square left, so its queen takes row 3, column 2; mint has one square left, so its queen takes row 4, column 11; khaki has one square left, so its queen takes row 8, column 1. 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 8, column 1.
The board after step 27 — 11 of 11 queens placed.
That completes the board: 11 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 11x11 Queens puzzle #238
Board #238 of the 11x11 bank carries a hard rating. Its 121 squares are shared between eleven colored regions, and the solution places eleven queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #238 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #238 are: pale yellow (1 square), teal (1 square), amber (4 squares), indigo (7 squares), mint (7 squares), khaki (8 squares), orange (13 squares), slate (18 squares), red (19 squares), blue (20 squares), sea green (23 squares). An even split would hand every color 11 squares, so the pale yellow region is running 10 squares under that average while the sea green region is the roomiest area on the grid with 23 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 pale yellow, with 1 square, is under far more pressure than sea green with 23. A 22-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Pale yellow is the tidiest shape on #238, packing 100% of its 1×1 bounding box, whereas orange straggles across its 4×7 box at just 46%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #238 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.
Pale yellow covers 1 square, spanning 1 row and 1 column (rows 7–7, columns 6–6). It never leaves row 7, so its queen is locked to that row.
Teal occupies 1 square within rows 11–11, columns 10–10. It never leaves row 11, so its queen is locked to that row.
Amber: 4 squares across rows 1–2, columns 9–11.
Indigo covers 7 squares, spanning 2 rows and 4 columns (rows 1–2, columns 5–8).
Mint occupies 7 squares within rows 2–5, columns 10–11.
Khaki: 8 squares across rows 4–10, columns 1–2.
Orange covers 13 squares, spanning 4 rows and 7 columns (rows 8–11, columns 1–7).
Slate occupies 18 squares within rows 3–8, columns 7–11.
Red: 19 squares across rows 3–9, columns 3–7.
Blue covers 20 squares, spanning 6 rows and 5 columns (rows 1–6, columns 1–5).
Sea green occupies 23 squares within rows 7–11, columns 5–11.
Where to start on this board
Begin where the choices are fewest. On this board that means pale yellow, which owns just 1 square inside a 1×1 window running from row 7 to row 7 and column 6 to column 6. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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. Teal, indigo and amber 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 #238
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 10 and 11 are the narrowest, touched by only 3 colors. Columns 1, 2, 3, 4, 8 and 9 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 10 rows. Row 8, meeting 5 colors, is the opposite case — the busiest line on #238 and the last one you should expect to crack.
Working through the middle game
The middle of #238 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
Keep alternating between two views of the board. In the region view you ask which of the eleven 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 11x11 board unwinds.
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 #238 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers four 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.
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.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
Techniques that work well at 11x11
At 11x11 you are managing 121 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
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 #238, start hunting for it near row 1.
Here is a shape check that gives nothing away: when #238 is solved, 4 queens sit on the outer edge of the grid and 7 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 11x11 Queens puzzle #238?
Puzzle #238 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #238 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #238, has been verified to have exactly one valid arrangement of eleven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #238?
Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.