11x11 Queens Puzzle #12
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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #12
What follows is a complete solve of puzzle #12, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 11 queen placements and 7 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 columns.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, line confined to one region, pairing argument on columns, last square in a column.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Only row 11, column 11 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 11, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
Step 3. The mint region has 4 squares left, spread across rows 4 and 5, columns 1–3. 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.
The board after step 3 — 1 of 11 queens placed. Step 4. All of the slate region's remaining squares (2 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.
Step 5. Sea green is down to 5 candidates in rows 8–10, columns 6–8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.
Step 6. Here is the pairing argument. The amber, blue, mint, and slate regions have no open squares outside rows 2–5 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 17 squares come off in one stroke.
The board after step 6 — 1 of 11 queens placed. Step 7. Column 9 has only pale yellow squares left (5 open squares). The column needs a queen and only pale yellow can provide one, which pins that colour to this column and kills its squares elsewhere: 2 squares.
Step 8. Here is the pairing argument. The amber, mint, and orange regions have no open squares outside columns 1–3 — 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: 3 squares come off in one stroke.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 4; 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 10; mint has one square left, so its queen takes row 4, column 3; amber has one square left, so its queen takes row 2, column 1; red has one square left, so its queen takes row 6, column 7; column 8 is down to one square, putting a queen on row 10, column 8; khaki has one square left, so its queen takes row 8, column 6; pale yellow has one square left, so its queen takes row 7, column 9; 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.
The board after step 9 — 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 #12
This is puzzle number 12 in the 11x11 collection, rated hard. The grid holds 121 squares split between eleven colored regions, and a finished board carries eleven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #12, 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 #12 are: teal (1 square), indigo (2 squares), mint (4 squares), amber (7 squares), slate (9 squares), sea green (9 squares), blue (14 squares), khaki (14 squares), orange (18 squares), red (21 squares), pale yellow (22 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the pale yellow region is the roomiest area on the grid with 22 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Teal is working with 1 square while pale yellow has 22 to spare, a gap of 21. Attack the cramped end and the roomy regions tend to resolve themselves.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×2 box that encloses it, while blue is the most scattered at 35% of its 5×8 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
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Teal: 1 square across rows 11–11, columns 11–11. It never leaves row 11, so its queen is locked to that row.
Indigo covers 2 squares, spanning 1 row and 2 columns (rows 1–1, columns 3–4). It never leaves row 1, so its queen is locked to that row.
Mint occupies 4 squares within rows 4–5, columns 1–3.
Amber: 7 squares across rows 1–4, columns 1–2.
Slate covers 9 squares, spanning 6 rows and 2 columns (rows 1–6, columns 10–11).
Sea green occupies 9 squares within rows 8–11, columns 6–9.
Blue: 14 squares across rows 1–5, columns 2–9.
Khaki covers 14 squares, spanning 6 rows and 5 columns (rows 6–11, columns 4–8).
Orange occupies 18 squares within rows 6–11, columns 1–3.
Red: 21 squares across rows 2–8, columns 4–10.
Pale yellow covers 22 squares, spanning 9 rows and 4 columns (rows 3–11, columns 8–11).
Where to start on this board
Open on the tightest color. Here that is teal, confined to 1 square between rows 11 and 11 and columns 11 and 11. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #12.
Because teal never leaves row 11, its queen must be in that row — and that instantly settles row 11 for everyone else. Sweep along it and X out every square that is not teal. 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 mint 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 #12
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 7, 9 and 10 are the narrowest, touched by only 4 colors. Columns 1, 5, 10 and 11 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 4 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 4 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.
With eleven queens to seat and eleven regions competing over 121 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 11x11 boards quickly and players who stall halfway.
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 #12 is rated hard
Ratings track how generous the opening is. On #12 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 4 colors, and a 21-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
Stuck on #12 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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around column 1 on this board.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 8 queens in the middle, including 1 in a corner. 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 11x11 Queens puzzle #12?
Puzzle #12 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 4 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #12 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #12, 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 #12?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.