12x12 Queens Puzzle #60
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 25-step reasoning is written out below.
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Step-by-step solution for puzzle #60
What follows is a complete solve of puzzle #60, worked the way a person works it rather than the way a computer does. It runs to 25 deductions — 12 queen placements and 13 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 pairing argument on columns.
Techniques this board needs: region shadow, region confined to one line, pairing argument on columns, last square in a region, pairing argument on rows, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Wherever indigo ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 2. All of the pale yellow region's remaining squares (2 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 14 squares come off.
Step 3. The slate region has 7 squares left, spread across rows 5–10, columns 11 and 12. 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. Wherever khaki 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.
The board after step 4 — 0 of 12 queens placed. Step 5. Rose cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to rose, so every square in it owned by another colour is dead: 10 squares in all.
Step 6. All of the orange region's remaining squares (3 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 8 squares come off.
Step 7. Amber, slate, and teal are now boxed into columns 9, 11, and 12 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 9 squares at once.
Step 8. The khaki region is down to one open square, at row 8, column 8. Every colour holds exactly one queen, so that square is forced. Row 8 and column 8 are now spoken for, along with the squares touching that queen.
Step 9. Only row 7, column 10 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 10, the rest of the pale yellow region and every square adjacent to it — diagonals included.
The board after step 9 — 2 of 12 queens placed. Step 10. Wherever slate 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. Orange cannot leave column 2 — 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 12. Here is the pairing argument. The sea green, teal, and rose regions have no open squares outside rows 10–12 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 13. 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: orange has one square left, so its queen takes row 9, column 2; red has one square left, so its queen takes row 6, column 6; slate has one square left, so its queen takes row 5, column 11; row 10 is down to one square, putting a queen on row 10, 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 10, column 7.
The board after step 13 — 6 of 12 queens placed. Step 14. Blue cannot leave column 5 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 4 squares in all.
Step 15. 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: 1 square come off.
Step 16. Wherever indigo ends up among its 2 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 17. 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: blue has one square left, so its queen takes row 3, column 5; mint has one square left, so its queen takes row 4, column 3; indigo has one square left, so its queen takes row 1, column 4; amber has one square left, so its queen takes row 2, column 12; teal has one square left, so its queen takes row 11, column 9; rose has one square left, so its queen takes row 12, 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 12, column 1.
The board after step 17 — 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 #60
This is puzzle number 60 in the 12x12 collection, rated medium. The grid holds 144 squares split between twelve colored regions, and a finished board carries twelve queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #60, 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 #60 are: pale yellow (2 squares), rose (3 squares), indigo (5 squares), khaki (6 squares), orange (8 squares), teal (9 squares), amber (12 squares), slate (12 squares), sea green (14 squares), blue (15 squares), mint (24 squares), red (34 squares). An even split would hand every color 12 squares, so the pale yellow region is running 10 squares under that average while the red region is the roomiest area on the grid with 34 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Pale yellow is working with 2 squares while red has 34 to spare, a gap of 32. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Pale yellow is the tidiest shape on #60, packing 100% of its 2×1 bounding box, whereas red straggles across its 11×9 box at just 34%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #60 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Pale yellow: 2 squares across rows 7–8, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.
Rose covers 3 squares, spanning 3 rows and 1 column (rows 10–12, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Indigo occupies 5 squares within rows 1–3, columns 3–5.
Khaki: 6 squares across rows 8–10, columns 8–11.
Orange covers 8 squares, spanning 6 rows and 2 columns (rows 6–11, columns 1–2).
Teal occupies 9 squares within rows 10–12, columns 9–12.
Amber: 12 squares across rows 1–5, columns 9–12.
Slate covers 12 squares, spanning 8 rows and 3 columns (rows 3–10, columns 10–12).
Sea green occupies 14 squares within rows 8–12, columns 5–9.
Blue: 15 squares across rows 1–6, columns 5–8.
Mint covers 24 squares, spanning 6 rows and 6 columns (rows 1–6, columns 1–6).
Red occupies 34 squares within rows 2–12, columns 2–10.
Where to start on this board
The most constrained color is where a solve should begin, and on #60 that is pale yellow with 2 squares. It reaches from row 7 to row 8 and from column 10 to column 10 — 2 rows by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because pale yellow stays inside column 10, its queen belongs to that column and no other region may use it. Cross out the rest of column 10 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
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 #60
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 7 and 12 are the narrowest, touched by only 4 colors. Columns 1, 2, 3, 4, 7 and 12 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 4 colors, so one of that handful must take the row and the rest are squeezed into the other 11 rows. Row 10 sits at the far end with 7 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.
Two checks, run alternately, will carry you through the 144 squares of this grid. First: has any of the twelve regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 12x12 boards almost always run one of those checks and forget the other.
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 #60 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 4 colors, and a region-size spread of 32. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
Stuck on #60 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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around column 1 on this board.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 9 queens in the middle, including 1 in a corner. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #60?
Puzzle #60 is rated medium. Its smallest color region holds 2 squares, 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 #60 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #60, 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 #60?
Start with the pale yellow region — the most constrained color on this board, with 2 squares spanning rows 7–8 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.