12x12 Queens Puzzle #226
Difficulty: hard. Place 12 queens on this 12x12 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 29-step reasoning is written out below.
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Step-by-step solution for puzzle #226
What follows is a complete solve of puzzle #226, worked the way a person works it rather than the way a computer does. It runs to 29 deductions — 12 queen placements and 17 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, line confined to one region.
Show the full walkthrough (26 steps) — contains the solution
Step 1. Only row 3, column 5 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 5, the rest of the blue region and every square adjacent to it — diagonals included.
Step 2. The rose region is down to one open square, at row 12, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 12 and column 10, the remaining rose squares, and the ring of neighbours around row 12, column 10.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 9 squares in all.
Step 4. The amber region is down to one open square, at row 2, column 12. Every colour holds exactly one queen, so that square is forced. Row 2 and column 12 are now spoken for, along with the squares touching that queen.
Step 5. Only row 4, column 9 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 9, the rest of the mint region and every square adjacent to it — diagonals included.
Step 6. The pale yellow region has 7 squares left, spread across rows 6–11, 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. Khaki cannot leave column 6 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 6 squares in all.
The board after step 7 — 4 of 12 queens placed. Step 8. The indigo region is down to one open square, at row 1, column 7. Every colour holds exactly one queen, so that square is forced. Row 1 and column 7 are now spoken for, along with the squares touching that queen.
Step 9. The red region has 2 squares left, spread across rows 5 and 6, columns 8 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 10. Wherever orange ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 11. Pale yellow is down to 6 candidates in rows 6–11, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 12. Here is the pairing argument. The red and sea green regions have no open squares outside columns 8 and 11 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 1 square come off in one stroke.
Step 13. Test row 5, column 1 for slate: place a queen there, run the forced consequences through, and row 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 13 — 5 of 12 queens placed. Step 14. Every open square left in column 1 (4 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, 2 squares in all.
Step 15. Test row 5, column 3 for slate: place a queen there, run the forced consequences through, and the orange 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 slate queen went to row 5, column 4. Follow the forced moves that would create and column 2 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. Test row 6, column 2 for slate: place a queen there, run the forced consequences through, and the teal 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 slate queen went to row 6, column 3. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Test row 6, column 4 for slate: place a queen there, run the forced consequences through, and column 2 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 20. Now a short proof by contradiction. Suppose the slate queen went to row 7, column 2. Follow the forced moves that would create and column 11 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 20 — 5 of 12 queens placed. Step 21. Test row 7, column 4 for slate: 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. The slate region is down to one open square, at row 5, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 2, the remaining slate squares, and the ring of neighbours around row 5, column 2.
Step 23. Only row 6, column 8 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 8, in red and in the ring around that square is now dead.
Step 24. The khaki region is down to one open square, at row 8, column 6. Every colour holds exactly one queen, so that square is forced. Row 8 and column 6 are now spoken for, along with the squares touching that queen.
Step 25. Orange cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 3 squares in all.
Step 26. 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: teal has one square left, so its queen takes row 11, column 4; orange has one square left, so its queen takes row 9, column 3; pale yellow has one square left, so its queen takes row 7, column 1; sea green has one square left, so its queen takes row 10, column 11. 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 11.
The board after step 26 — 12 of 12 queens placed.
That completes the board: 12 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 12x12 Queens puzzle #226
Board #226 of the 12x12 bank carries a hard rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve 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, #226 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 #226 are: blue (1 square), rose (1 square), indigo (4 squares), orange (4 squares), amber (5 squares), pale yellow (7 squares), mint (11 squares), khaki (12 squares), sea green (22 squares), red (25 squares), teal (25 squares), slate (27 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the slate region is the roomiest area on the grid with 27 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Blue has only 1 place to put its queen; slate has 27. The spread between them on this board is 26 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. Blue is the densest region on this grid at 100% of a 1×1 frame; red is the loosest, occupying only 51% of its 7×7 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.
Blue covers 1 square, spanning 1 row and 1 column (rows 3–3, columns 5–5). It never leaves row 3, so its queen is locked to that row.
Rose occupies 1 square within rows 12–12, columns 10–10. It never leaves row 12, so its queen is locked to that row.
Indigo: 4 squares across rows 1–2, columns 5–7.
Orange covers 4 squares, spanning 3 rows and 2 columns (rows 8–10, columns 2–3).
Amber occupies 5 squares within rows 1–3, columns 11–12.
Pale yellow: 7 squares across rows 6–11, columns 1–2.
Mint covers 11 squares, spanning 4 rows and 4 columns (rows 1–4, columns 8–11).
Khaki occupies 12 squares within rows 2–8, columns 5–7.
Sea green: 22 squares across rows 6–12, columns 8–12.
Red covers 25 squares, spanning 7 rows and 7 columns (rows 3–9, columns 6–12).
Teal occupies 25 squares within rows 8–12, columns 1–9.
Slate: 27 squares across rows 1–7, columns 1–5.
Where to start on this board
Blue is the natural opening on #226: it is the smallest region on the grid at 1 square, boxed into rows 3–3 and columns 5–5. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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. Rose and indigo 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 #226
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 5, 11 and 12 are the narrowest, touched by only 3 colors. Column 4 is the tightest column, with 2 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 11 rows. Row 8, meeting 6 colors, is the opposite case — the busiest line on #226 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 twelve 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 12x12 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 #226 is rated hard
The hard label comes out of four measurable things about #226: four 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 26. 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.
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 12x12
At 12x12 you are managing 144 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 #226 the place to look for one is around column 4.
Here is a shape check that gives nothing away: when #226 is solved, 4 queens sit on the outer edge of the grid and 8 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 12x12 Queens puzzle #226?
Puzzle #226 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 12x12 puzzle #226 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #226, has been verified to have exactly one valid arrangement of twelve queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #226?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.