12x12 Queens Puzzle #54

Difficulty: easy. 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 44-step reasoning is written out below.

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Step-by-step solution for puzzle #54

What follows is a complete solve of puzzle #54, worked the way a person works it rather than the way a computer does. It runs to 44 deductions — 12 queen placements and 32 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, proof by contradiction, pairing argument on columns, pairing argument on rows.

Show the full walkthrough (36 steps) — contains the solution

Step 1. Only row 1, column 12 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 12, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. All of the mint region's remaining squares (2 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 13 squares come off.
Step 3. The orange region has 4 squares left, spread across rows 9–11, columns 5 and 6. 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 rose 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 5. Orange is down to 3 candidates in rows 9 and 10, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 6. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 1. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 7. Test row 2, column 2 for amber: place a queen there, run the forced consequences through, and the rose region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 5. Follow the forced moves that would create and row 11 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 9. Test row 2, column 6 for amber: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 9 — 1 of 12 queens placed.
Step 10. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 7. Follow the forced moves that would create and the sea green region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 11. Test row 3, column 3 for amber: place a queen there, run the forced consequences through, and the rose region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 5. Follow the forced moves that would create and row 11 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 13. Test row 3, column 6 for amber: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 5, column 5. Follow the forced moves that would create and row 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Test row 5, column 6 for amber: place a queen there, run the forced consequences through, and the slate 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 amber queen went to row 5, column 7. Follow the forced moves that would create and row 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 17. Test row 6, column 6 for amber: place a queen there, run the forced consequences through, and the pale yellow 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 2, column 8. Follow the forced moves that would create and the teal region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
The board after step 18 — 1 of 12 queens placed.
Step 19. Test row 3, column 7 for blue: place a queen there, run the forced consequences through, and column 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 20. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 8. Follow the forced moves that would create and the teal region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 21. Test row 5, column 10 for blue: place a queen there, run the forced consequences through, and row 2 runs out of open squares entirely. 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 11. Follow the forced moves that would create and row 2 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 23. Test row 6, column 10 for blue: place a queen there, run the forced consequences through, and row 2 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 24. Now a short proof by contradiction. Suppose the blue queen went to row 6, column 11. Follow the forced moves that would create and row 2 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 25. Test row 4, column 1 for mint: place a queen there, run the forced consequences through, and the slate region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 26. The mint region is down to one open square, at row 4, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 2, the remaining mint squares, and the ring of neighbours around row 4, column 2.
Step 27. Test row 2, column 4 for amber: place a queen there, run the forced consequences through, and the sea green region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 27 — 2 of 12 queens placed.
Step 28. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 4. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 29. Only row 2, column 3 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 3, the rest of the amber region and every square adjacent to it — diagonals included.
Step 30. All of the red region's remaining squares (5 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 31. Slate, orange, and teal are now boxed into columns 4–6 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 3 squares at once.
Step 32. All of the rose region's remaining squares (2 open squares) lie in column 7. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 33. Sea green cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 6 squares in all.
Step 34. Here is the pairing argument. The orange and sea green regions have no open squares outside rows 9 and 10 — 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: 8 squares come off in one stroke.
Step 35. Teal and rose are now boxed into rows 11 and 12 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 6 squares at once.
Step 36. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: khaki has one square left, so its queen takes row 8, column 9; pale yellow has one square left, so its queen takes row 7, column 11; blue has one square left, so its queen takes row 3, column 10; red has one square left, so its queen takes row 6, column 1; slate has one square left, so its queen takes row 5, column 4; sea green has one square left, so its queen takes row 10, column 8; teal has one square left, so its queen takes row 11, column 5; orange has one square left, so its queen takes row 9, column 6; rose has one square left, so its queen takes row 12, column 7. 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 12, column 7.
The board after step 36 — 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 #54

Board #54 of the 12x12 bank carries a easy 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, #54 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 #54 are: indigo (1 square), mint (2 squares), orange (4 squares), rose (4 squares), sea green (10 squares), teal (10 squares), red (11 squares), pale yellow (14 squares), slate (18 squares), khaki (19 squares), blue (25 squares), amber (26 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the amber region is the roomiest area on the grid with 26 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 indigo, with 1 square, is under far more pressure than amber with 26. A 25-square gap like this one rewards players who start where the choices are fewest.

A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and sea green the most diffuse at 42% of its 6×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

Here is the full footprint of every color on puzzle #54, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Indigo: 1 square across rows 1–1, columns 12–12. It never leaves row 1, so its queen is locked to that row.

Mint covers 2 squares, spanning 1 row and 2 columns (rows 4–4, columns 1–2). It never leaves row 4, so its queen is locked to that row.

Orange occupies 4 squares within rows 9–11, columns 5–6.

Rose: 4 squares across rows 11–12, columns 5–7.

Sea green covers 10 squares, spanning 6 rows and 4 columns (rows 6–11, columns 5–8).

Teal occupies 10 squares within rows 10–12, columns 1–5.

Red: 11 squares across rows 5–11, columns 1–3.

Pale yellow covers 14 squares, spanning 4 rows and 5 columns (rows 5–8, columns 8–12).

Slate occupies 18 squares within rows 4–9, columns 2–6.

Khaki: 19 squares across rows 8–12, columns 8–12.

Blue covers 25 squares, spanning 6 rows and 6 columns (rows 1–6, columns 7–12).

Amber occupies 26 squares within rows 1–6, columns 1–7.

Where to start on this board

Indigo is the natural opening on #54: it is the smallest region on the grid at 1 square, boxed into rows 1–1 and columns 12–12. 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 indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Mint and rose 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 #54

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2 and 3 are the narrowest, touched by only 2 colors. Columns 4, 9, 10 and 11 are the tightest columns, with 3 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 11 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

Working through the middle game

The middle of #54 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 twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 solve from a stalled one.

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 #54 is rated easy

The rating reflects how much the board concedes up front. Puzzle #54 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 25 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.

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.

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 #54, start hunting for it near row 2.

As a sanity check rather than a spoiler: the finished solution to #54 puts 3 queens on the outer border and 9 queens in the interior, 1 of them in an actual corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

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 #54?

Puzzle #54 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #54 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #54, 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 #54?

Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 12–12. Narrowing the smallest region first removes the most squares elsewhere on the grid.