12x12 Queens Puzzle #170

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 27-step reasoning is written out below.

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Step-by-step solution for puzzle #170

What follows is a complete solve of puzzle #170, worked the way a person works it rather than the way a computer does. It runs to 27 deductions — 12 queen placements and 15 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 shadow, proof by contradiction, last square in a region, region confined to one line, pairing argument on rows.

Show the full walkthrough (19 steps) — contains the solution

  1. Step 1. Wherever slate ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.

  2. Step 2. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 7. Follow the forced moves that would create and the blue region is left with nowhere to put its queen — impossible. So that square is ruled out for good.

  3. Step 3. Test row 1, column 8 for indigo: 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.

  4. Step 4. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 9. 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.

  5. Step 5. Test row 2, column 7 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.

    Puzzle #170 after step 5: 0 of 12 queens placed
    The board after step 5 — 0 of 12 queens placed.

  6. Step 6. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 9. 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.

  7. Step 7. Test row 3, column 9 for indigo: 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.

  8. Step 8. Indigo is down to 3 candidates in rows 1, 4, and 5, columns 8–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.

  9. Step 9. Test row 4, column 9 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.

  10. Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 5, 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.

    Puzzle #170 after step 10: 0 of 12 queens placed
    The board after step 10 — 0 of 12 queens placed.

  11. Step 11. Only row 1, column 10 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 10, in indigo and in the ring around that square is now dead.

  12. Step 12. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.

  13. Step 13. Only row 3, column 12 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 12, the rest of the blue region and every square adjacent to it — diagonals included.

  14. Step 14. Mint is down to 3 candidates in rows 4 and 5, columns 4 and 5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about mint itself.

    Puzzle #170 after step 14: 2 of 12 queens placed
    The board after step 14 — 2 of 12 queens placed.

  15. Step 15. Slate cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 7 squares in all.

  16. Step 16. Here is the pairing argument. The mint and slate regions have no open squares outside rows 4 and 5 — 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.

  17. Step 17. 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: red has one square left, so its queen takes row 6, column 2; amber has one square left, so its queen takes row 2, column 1; pale yellow has one square left, so its queen takes row 7, column 9; khaki has one square left, so its queen takes row 8, column 3. 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 8, column 3.

  18. Step 18. Orange cannot leave column 5 — 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: 7 squares in all.

  19. Step 19. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: mint has one square left, so its queen takes row 4, column 4; slate has one square left, so its queen takes row 5, column 7; teal has one square left, so its queen takes row 11, column 6; orange has one square left, so its queen takes row 9, column 5; sea green has one square left, so its queen takes row 10, column 8; rose has one square left, so its queen takes row 12, 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 12, column 11.

    Puzzle #170 after step 19: 12 of 12 queens placed
    The board after step 19 — 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 #170

Board #170 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.

There is exactly one way to finish #170, 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 #170 are: slate (3 squares), amber (4 squares), blue (7 squares), teal (7 squares), indigo (10 squares), red (10 squares), rose (13 squares), khaki (15 squares), mint (16 squares), pale yellow (17 squares), orange (18 squares), sea green (24 squares). An even split would hand every color 12 squares, so the slate region is running 9 squares under that average while the sea green region is the roomiest area on the grid with 24 squares.

Every color takes one queen and one only, whatever its size — which is why the 21 square difference between slate and sea green is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 2×2 frame; pale yellow is the loosest, occupying only 49% of its 5×7 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.

Slate occupies 3 squares within rows 4–5, columns 6–7.

Amber: 4 squares across rows 1–2, columns 1–2.

Blue covers 7 squares, spanning 3 rows and 3 columns (rows 1–3, columns 10–12).

Teal occupies 7 squares within rows 10–12, columns 3–6.

Indigo: 10 squares across rows 1–5, columns 7–10.

Red covers 10 squares, spanning 4 rows and 4 columns (rows 3–6, columns 1–4).

Rose occupies 13 squares within rows 9–12, columns 8–12.

Khaki: 15 squares across rows 5–8, columns 1–7.

Mint covers 16 squares, spanning 5 rows and 6 columns (rows 1–5, columns 3–8).

Pale yellow occupies 17 squares within rows 3–7, columns 6–12.

Orange: 18 squares across rows 8–12, columns 1–6.

Sea green covers 24 squares, spanning 6 rows and 7 columns (rows 7–12, columns 6–12).

Where to start on this board

Slate is the natural opening on #170: it is the smallest region on the grid at 3 squares, boxed into rows 4–5 and columns 6–7. Since one of those squares must hold a queen, that block of 2 rows and 2 columns is the most contested part of the board.

With 2 rows and 2 columns of room, slate needs narrowing rather than solving outright. Test its 3 squares against the neighbours: a candidate that would strand another color is dead, and eliminating even two of them tightens the whole corner of the board around rows 4–5.

Work outward to the other flat regions next. Amber 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 #170

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6, 7, 8 and 9 are the narrowest, touched by only 3 colors. Columns 1, 2, 5, 9, 11 and 12 are the tightest columns, with 4 colors.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 5 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

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

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 #170 is rated hard

The hard label comes out of four measurable things about #170: it has no region locked to one row or column, its smallest color spans 3 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 21. Boards that score tight on those measures give up their first queen fast.

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.

Stuck on #170 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 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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #170 is row 6.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 8 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.

How to play Queens

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. 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.
  5. 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 #170?

Puzzle #170 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.

Does 12x12 puzzle #170 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #170, 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 #170?

Start with the slate region — the most constrained color on this board, with 3 squares spanning rows 4–5 and columns 6–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.