12x12 Queens Puzzle #157
Difficulty: medium. 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 #157
What follows is a complete solve of puzzle #157, 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, proof by contradiction, pairing argument on rows, pairing argument on columns.
Show the full walkthrough (25 steps) — contains the solution
Step 1. Only row 1, column 11 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 11, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. All of the orange region's remaining squares (4 open squares) lie in column 12. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 6 squares come off.
Step 3. The blue region has 3 squares left, spread across rows 2 and 3, columns 8–10. 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. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 5. Follow the forced moves that would create and the pale yellow region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 5. Test row 3, column 5 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 6. 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 rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
The board after step 6 — 1 of 12 queens placed. Step 7. Wherever amber 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 8. 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 rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 9. Test row 4, column 7 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 10. Wherever amber ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 11. Test row 2, column 8 for blue: 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 12. The blue region has 2 squares left, spread across rows 2 and 3, columns 9 and 10. 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 13. Test row 2, column 9 for blue: 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 13 — 1 of 12 queens placed. Step 14. The blue region is down to one open square, at row 3, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 10, the remaining blue squares, and the ring of neighbours around row 3, column 10.
Step 15. The red region has 6 squares left, spread across rows 5, 6, 8, and 9, 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 16. Here is the pairing argument. The amber and mint regions have no open squares outside rows 2 and 4 — 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: 6 squares come off in one stroke.
Step 17. Mint, red, sea green, and rose are now boxed into columns 6–9 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 9 squares at once.
Step 18. The pale yellow region is down to one open square, at row 7, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 5, the remaining pale yellow squares, and the ring of neighbours around row 7, column 5.
Step 19. Only row 2, column 4 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 4, in amber and in the ring around that square is now dead.
The board after step 19 — 4 of 12 queens placed. Step 20. The mint region is down to one open square, at row 4, column 8. Every colour holds exactly one queen, so that square is forced. Row 4 and column 8 are now spoken for, along with the squares touching that queen.
Step 21. Red cannot leave column 9 — every one of its open squares (3 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 3 squares in all.
Step 22. The sea green region is down to one open square, at row 10, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 7, the remaining sea green squares, and the ring of neighbours around row 10, column 7.
Step 23. Only row 12, column 6 is still open in rose, and since each colour must take a queen, that settles it. Everything else in row 12, in column 6, in rose and in the ring around that square is now dead.
Step 24. All of the teal region's remaining squares (2 open squares) lie in row 11. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 25. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: orange has one square left, so its queen takes row 9, column 12; khaki has one square left, so its queen takes row 8, column 3; red has one square left, so its queen takes row 6, column 9; teal has one square left, so its queen takes row 11, column 2; slate has one square left, so its queen takes row 5, column 1. 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 5, column 1.
The board after step 25 — 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 #157
Puzzle #157 is one of the medium-rated boards in the 12x12 set. Twelve colors divide its 144 squares, and you win by placing twelve queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
There is exactly one way to finish #157, 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 twelve regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #157 are: indigo (1 square), rose (7 squares), pale yellow (8 squares), amber (10 squares), mint (10 squares), red (10 squares), orange (10 squares), teal (12 squares), sea green (13 squares), khaki (14 squares), blue (20 squares), slate (29 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the slate region is the roomiest area on the grid with 29 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 slate with 29. A 28-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #157, packing 100% of its 1×1 bounding box, whereas mint straggles across its 5×5 box at just 40%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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 medium board.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 11–11). It never leaves row 1, so its queen is locked to that row.
Rose occupies 7 squares within rows 10–12, columns 6–9.
Pale yellow: 8 squares across rows 5–9, columns 5–7.
Amber covers 10 squares, spanning 4 rows and 4 columns (rows 1–4, columns 4–7).
Mint occupies 10 squares within rows 1–5, columns 6–10.
Red: 10 squares across rows 5–9, columns 7–10.
Orange covers 10 squares, spanning 6 rows and 2 columns (rows 7–12, columns 11–12).
Teal occupies 12 squares within rows 10–12, columns 1–5.
Sea green: 13 squares across rows 7–12, columns 7–10.
Khaki covers 14 squares, spanning 5 rows and 6 columns (rows 7–11, columns 1–6).
Blue occupies 20 squares within rows 1–8, columns 8–12.
Slate: 29 squares across rows 1–8, columns 1–5.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 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 #157.
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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: orange spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #157
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 3, 4, 6 and 12 are the narrowest, touched by only 4 colors. Column 12 is the tightest column, with 2 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 4 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 8 meets 7 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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.
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 #157 is rated medium
The medium label comes out of four measurable things about #157: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 4 colors, and its largest and smallest regions differ by 28. Boards that score tight on those measures give up their first queen fast.
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 #157 despite the medium 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 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 #157, start hunting for it near column 12.
As a sanity check rather than a spoiler: the finished solution to #157 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 #157?
Puzzle #157 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 4 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #157 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #157, 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 #157?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.