12x12 Queens Puzzle #8
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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #8
What follows is a complete solve of puzzle #8, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 12 queen placements and 12 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 rows.
Techniques this board needs: region shadow, region confined to one line, last square in a region, pairing argument on rows, line confined to one region, last square in a column, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Wherever teal 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 2. 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: 12 squares come off.
Step 3. Only row 11, column 1 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 1, in teal and in the ring around that square is now dead.
Step 4. All of the indigo region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.
The board after step 4 — 1 of 12 queens placed. Step 5. Only row 3, column 3 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 3, the rest of the blue region and every square adjacent to it — diagonals included.
Step 6. The indigo region is down to one open square, at row 1, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 2, the remaining indigo squares, and the ring of neighbours around row 1, column 2.
Step 7. 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 8. Sea green is down to 3 candidates in rows 9 and 10, columns 10 and 12, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about sea green itself.
Step 9. Orange and sea green are now boxed into rows 9 and 10 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 8 squares at once.
The board after step 9 — 3 of 12 queens placed. Step 10. Wherever khaki 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 11. Column 7 has only red squares left (5 open squares). The column needs a queen and only red can provide one, which pins that colour to this column and kills its squares elsewhere: 14 squares.
Step 12. Every open square left in column 5 (2 open squares) belongs to the amber region. Since that column must contain a queen, amber is the colour that supplies it — so amber squares anywhere else on the board are out, 6 squares in all.
Step 13. Column 4 has only orange squares left (2 open squares). The column needs a queen and only orange can provide one, which pins that colour to this column and kills its squares elsewhere: 1 square.
The board after step 13 — 3 of 12 queens placed. Step 14. Column 6 now has exactly one open square, row 8, column 6, in the khaki region. Columns must each hold a queen, so that placement is forced. That clears row 8, column 6, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 15. Row 7 has only pale yellow squares left (4 open squares). The row needs a queen and only pale yellow can provide one, which pins that colour to this row and kills its squares elsewhere: 3 squares.
Step 16. Every open square left in column 8 (2 open squares) belongs to the mint region. Since that column must contain a queen, mint is the colour that supplies it — so mint squares anywhere else on the board are out, 4 squares in all.
Step 17. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: row 6 is down to one square, putting a queen on row 6, column 7; mint has one square left, so its queen takes row 4, column 8; amber has one square left, so its queen takes row 2, column 5; slate has one square left, so its queen takes row 5, column 12; sea green has one square left, so its queen takes row 10, column 10; orange has one square left, so its queen takes row 9, column 4; rose has one square left, so its queen takes row 12, column 9; pale yellow has one square left, so its queen takes row 7, 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 7, column 11.
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 #8
This is puzzle number 8 in the 12x12 collection, rated hard. 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 #8, 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 #8 are: rose (2 squares), teal (4 squares), indigo (5 squares), mint (6 squares), slate (6 squares), blue (8 squares), sea green (9 squares), pale yellow (12 squares), khaki (16 squares), amber (20 squares), orange (25 squares), red (31 squares). An even split would hand every color 12 squares, so the rose region is running 10 squares under that average while the red region is the roomiest area on the grid with 31 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Rose is working with 2 squares while red has 31 to spare, a gap of 29. Attack the cramped end and the roomy regions tend to resolve themselves.
Shape matters as much as size. The rose region is the most compact color here, filling 100% of the 1×2 box that encloses it, while red is the most scattered at 43% of its 8×9 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
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.
Rose occupies 2 squares within rows 12–12, columns 9–10. It never leaves row 12, so its queen is locked to that row.
Teal: 4 squares across rows 11–12, columns 1–3.
Indigo covers 5 squares, spanning 3 rows and 2 columns (rows 1–3, columns 1–2).
Mint occupies 6 squares within rows 4–6, columns 8–10.
Slate: 6 squares across rows 1–5, columns 11–12.
Blue covers 8 squares, spanning 4 rows and 3 columns (rows 3–6, columns 1–3).
Sea green occupies 9 squares within rows 9–12, columns 9–12.
Pale yellow: 12 squares across rows 5–10, columns 9–12.
Khaki covers 16 squares, spanning 5 rows and 6 columns (rows 7–11, columns 5–10).
Amber occupies 20 squares within rows 1–6, columns 3–8.
Orange: 25 squares across rows 7–12, columns 1–8.
Red covers 31 squares, spanning 8 rows and 9 columns (rows 1–8, columns 3–11).
Where to start on this board
The most constrained color is where a solve should begin, and on #8 that is rose with 2 squares. It reaches from row 12 to row 12 and from column 9 to column 10 — 1 row by 2 columns. 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 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. Teal 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 #8
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 7, 8, 9, 10, 11 and 12 are the narrowest, touched by only 4 colors. Columns 4 and 12 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 4 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 5 is the crowded extreme at 6 colors — worth revisiting later, not now.
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.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #8 is rated hard
The hard label comes out of four measurable things about #8: one of its regions is locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 4 colors, and its largest and smallest regions differ by 29. 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 #8 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.
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 4 on this board.
A non-spoiling distribution check: solved, #8 carries 4 queens on the perimeter against 8 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 #8?
Puzzle #8 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 4 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #8 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #8, 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 #8?
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.