12x12 Queens Puzzle #151

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 52-step reasoning is written out below.

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Step-by-step solution for puzzle #151

What follows is a complete solve of puzzle #151, worked the way a person works it rather than the way a computer does. It runs to 52 deductions — 12 queen placements and 40 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: region confined to one line, region shadow, proof by contradiction, last square in a region, line confined to one region, last square in a row, pairing argument on rows.

Show the full walkthrough (44 steps) — contains the solution

Step 1. Slate cannot leave column 1 — every one of its open squares (4 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 8 squares in all.
Step 2. Khaki is down to 3 candidates in rows 8 and 9, columns 11 and 12, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about khaki itself.
Step 3. The orange region has 7 squares left, spread across rows 9–12, columns 7–9. 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 rose region's remaining squares (2 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 11 squares come off.
Step 5. Test row 2, column 2 for indigo: 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 6. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 3. Follow the forced moves that would create and the red region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 7. Test row 2, column 4 for indigo: 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 8. Now a short proof by contradiction. Suppose the indigo queen went to row 3, column 2. 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 9. Test row 3, column 4 for indigo: 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 10. Wherever indigo ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Test row 3, column 5 for indigo: 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.
The board after step 11 — 0 of 12 queens placed.
Step 12. The indigo region has 3 squares left, spread across rows 1 and 2, columns 2, 3, and 5. 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 13. Test row 1, column 7 for amber: place a queen there, run the forced consequences through, and the mint region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 1, column 8. Follow the forced moves that would create and the mint region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 15. Test row 2, column 5 for indigo: place a queen there, run the forced consequences through, and the mint region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 16. 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: 4 squares come off.
Step 17. Test row 2, column 7 for amber: place a queen there, run the forced consequences through, and the mint 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 amber queen went to row 2, column 8. Follow the forced moves that would create and the mint region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Test row 2, column 9 for amber: 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 20. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 10. 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 2, column 11 for amber: place a queen there, run the forced consequences through, and the mint 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 amber queen went to row 3, column 6. 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.
The board after step 22 — 0 of 12 queens placed.
Step 23. Test row 3, column 7 for amber: 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 24. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 8. 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 25. Test row 3, column 9 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.
Step 26. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 10. 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 27. Test row 3, column 11 for amber: 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 28. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 3. Follow the forced moves that would create and the red region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 29. Only row 1, column 2 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 2, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 30. Now a short proof by contradiction. Suppose the amber queen went to row 4, column 6. 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 31. Test row 4, column 9 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.
Step 32. The amber region is down to one open square, at row 2, column 6. Every colour holds exactly one queen, so that square is forced. Row 2 and column 6 are now spoken for, along with the squares touching that queen.
Step 33. The teal region has 4 squares left, spread across rows 10 and 11, columns 3–5. 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.
The board after step 33 — 2 of 12 queens placed.
Step 34. 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 red region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 35. Test row 4, column 4 for blue: 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 36. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 5. Follow the forced moves that would create and column 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 37. Row 4 has only mint squares left (5 open squares). The row needs a queen and only mint can provide one, which pins that colour to this row and kills its squares elsewhere: 13 squares.
Step 38. Row 3 has a single open square left, at column 3 in the blue region. Every row needs a queen, so this one is forced. Row 3 and column 3 are now spoken for, along with the squares touching that queen.
Step 39. Teal cannot leave row 11 — every one of its open squares (2 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 5 squares in all.
Step 40. Wherever orange ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 41. Pale yellow, khaki, orange, and sea green are now boxed into rows 7–10 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 7 squares at once.
Step 42. All of the red 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: 3 squares come off.
Step 43. Orange cannot leave column 9 — 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: 2 squares in all.
Step 44. 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: rose has one square left, so its queen takes row 12, column 10; sea green has one square left, so its queen takes row 10, column 11; khaki has one square left, so its queen takes row 8, column 12; mint has one square left, so its queen takes row 4, column 8; red has one square left, so its queen takes row 6, column 7; slate has one square left, so its queen takes row 5, column 1; orange has one square left, so its queen takes row 9, column 9; pale yellow has one square left, so its queen takes row 7, column 4; teal has one square left, so its queen takes row 11, column 5. 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 11, column 5.
The board after step 44 — 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 #151

12x12 puzzle #151 is rated easy. Twelve colored regions cover its 144 squares, and your job is to seat twelve queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

As with every puzzle in this collection, #151 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 #151 are: rose (2 squares), khaki (3 squares), slate (4 squares), red (7 squares), orange (7 squares), sea green (12 squares), indigo (13 squares), blue (13 squares), teal (14 squares), mint (19 squares), amber (23 squares), pale yellow (27 squares). An even split would hand every color 12 squares, so the rose region is running 10 squares under that average while the pale yellow region is the roomiest area on the grid with 27 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 rose, with 2 squares, is under far more pressure than pale yellow with 27. 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 slate is the most solid, filling 100% of its 4×1 box, and blue the most diffuse at 52% of its 5×5 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 #151, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Rose covers 2 squares, spanning 1 row and 2 columns (rows 12–12, columns 9–10). It never leaves row 12, so its queen is locked to that row.

Khaki occupies 3 squares within rows 8–9, columns 11–12.

Slate: 4 squares across rows 5–8, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Red covers 7 squares, spanning 4 rows and 3 columns (rows 5–8, columns 7–9).

Orange occupies 7 squares within rows 9–12, columns 7–9.

Sea green: 12 squares across rows 7–12, columns 10–12.

Indigo covers 13 squares, spanning 4 rows and 5 columns (rows 1–4, columns 1–5).

Blue occupies 13 squares within rows 3–7, columns 2–6.

Teal: 14 squares across rows 10–12, columns 1–7.

Mint covers 19 squares, spanning 6 rows and 6 columns (rows 2–7, columns 7–12).

Amber occupies 23 squares within rows 1–4, columns 4–12.

Pale yellow: 27 squares across rows 5–10, columns 1–8.

Where to start on this board

Begin where the choices are fewest. On this board that means rose, which owns just 2 squares inside a 1×2 window running from row 12 to row 12 and column 9 to column 10. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because rose never leaves row 12, its queen must be in that row — and that instantly settles row 12 for everyone else. Sweep along it and X out every square that is not rose. 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. Khaki is no more than two rows tall, so its queen is 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 #151

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 1, 2, 3, 6, 10, 11 and 12 are the tightest columns, with 4 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 7 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

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 twelve queens to seat and twelve regions competing over 144 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 12x12 boards quickly and players who stall halfway.

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 #151 is rated easy

The rating reflects how much the board concedes up front. Puzzle #151 has two regions confined to a single row or column, its smallest color holds 2 squares, 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.

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.

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 #151, start hunting for it near row 1.

As a sanity check rather than a spoiler: the finished solution to #151 puts 4 queens on the outer border and 8 queens in the interior, with none in a 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 #151?

Puzzle #151 is rated easy. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #151 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #151, 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 #151?

Start with the rose region — the most constrained color on this board, with 2 squares spanning rows 12–12 and columns 9–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.