11x11 Queens Puzzle #22
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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #22
What follows is a complete solve of puzzle #22, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 11 queen placements and 11 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 shadow, region confined to one line, pairing argument on rows, last square in a row.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 8, column 6 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 6, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 2. Indigo is down to 5 candidates in rows 1 and 2, columns 2–5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. The amber region has 4 squares left, spread across rows 1–3, columns 8 and 9. 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 4. Wherever slate 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.
Step 5. Pale yellow cannot leave row 7 — every one of its open squares (2 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 8 squares in all.
The board after step 5 — 1 of 11 queens placed. Step 6. The slate region has 3 squares left, spread across rows 5 and 6, columns 1 and 2. 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 7. Wherever red ends up among its 5 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. Teal is down to 4 candidates in rows 9–11, columns 10 and 11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about teal itself.
Step 9. Indigo, amber, and blue are now boxed into rows 1–3 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 14 squares at once.
Step 10. Wherever red 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 11. Only row 7, column 9 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 9, in pale yellow and in the ring around that square is now dead.
The board after step 11 — 2 of 11 queens placed. Step 12. Row 3 has a single open square left, at column 5 in the blue region. Every row needs a queen, so this one is forced. Cross out the whole of row 3 and column 5, the remaining blue squares, and the ring of neighbours around row 3, column 5.
Step 13. Amber cannot leave column 8 — 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: 5 squares in all.
Step 14. The red region is down to one open square, at row 6, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 7, the remaining red squares, and the ring of neighbours around row 6, column 7.
Step 15. The mint region has 3 squares left, spread across rows 4 and 5, columns 2–4. 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 16. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: slate has one square left, so its queen takes row 5, column 1; mint has one square left, so its queen takes row 4, column 3; indigo has one square left, so its queen takes row 1, column 2; amber has one square left, so its queen takes row 2, column 8; orange has one square left, so its queen takes row 9, column 10; teal has one square left, so its queen takes row 11, column 11; sea green has one square left, so its queen takes row 10, column 4. 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 10, column 4.
The board after step 16 — 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 #22
Board #22 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.
There is exactly one way to finish #22, 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 eleven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #22 are: khaki (1 square), amber (4 squares), pale yellow (4 squares), teal (4 squares), indigo (5 squares), blue (7 squares), slate (7 squares), red (14 squares), mint (19 squares), sea green (27 squares), orange (29 squares). An even split would hand every color 11 squares, so the khaki region is running 10 squares under that average while the orange region is the roomiest area on the grid with 29 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Khaki has only 1 place to put its queen; orange has 29. The spread between them on this board is 28 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Look at arrangement as well as area. Pale yellow is the densest region on this grid at 100% of a 2×2 frame; red is the loosest, occupying only 40% of its 7×5 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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.
Khaki covers 1 square, spanning 1 row and 1 column (rows 8–8, columns 6–6). It never leaves row 8, so its queen is locked to that row.
Amber occupies 4 squares within rows 1–3, columns 8–9.
Pale yellow: 4 squares across rows 7–8, columns 8–9.
Teal covers 4 squares, spanning 3 rows and 2 columns (rows 9–11, columns 10–11).
Indigo occupies 5 squares within rows 1–2, columns 2–5.
Blue: 7 squares across rows 1–3, columns 4–7.
Slate covers 7 squares, spanning 4 rows and 2 columns (rows 5–8, columns 1–2).
Red occupies 14 squares within rows 2–8, columns 4–8.
Mint: 19 squares across rows 1–6, columns 1–7.
Sea green covers 27 squares, spanning 7 rows and 7 columns (rows 5–11, columns 1–7).
Orange occupies 29 squares within rows 1–11, columns 7–11.
Where to start on this board
Begin where the choices are fewest. On this board that means khaki, which owns just 1 square inside a 1×1 window running from row 8 to row 8 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 khaki never leaves row 8, its queen must be in that row — and that instantly settles row 8 for everyone else. Sweep along it and X out every square that is not khaki. 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 pale yellow 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 #22
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 4, 9, 10 and 11 are the narrowest, touched by only 3 colors. Columns 10 and 11 are the tightest columns, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 10 rows. Row 8 is the crowded extreme at 6 colors — worth revisiting later, not now.
Working through the middle game
The middle of #22 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.
The whole middle game reduces to two sweeps. Sweep the eleven colors looking for one with a single square left; sweep the eleven rows and eleven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 11x11 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 #22 is rated hard
The hard label comes out of four measurable things about #22: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 28. Boards that score tight on those measures give up their first queen fast.
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 hard board from the rest.
Stuck on #22 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 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.
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 #22 the place to look for one is around column 10.
A non-spoiling distribution check: solved, #22 carries 3 queens on the perimeter against 8 queens inside, 1 of them cornered. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 #22?
Puzzle #22 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 11x11 puzzle #22 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #22, 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 #22?
Start with the khaki region — the most constrained color on this board, with 1 square spanning rows 8–8 and columns 6–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.