12x12 Queens Puzzle #98
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 25-step reasoning is written out below.
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Step-by-step solution for puzzle #98
What follows is a complete solve of puzzle #98, 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, proof by contradiction, line confined to one region, last square in a column, last square in a row.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 12, column 3 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 3, the rest of the rose region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (4 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.
Step 3. The amber region has 3 squares left, spread across rows 2 and 3, columns 7 and 8. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 4. All of the indigo region's remaining squares (3 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 1 square come off.
Step 5. Pale yellow is down to 4 candidates in rows 5–7, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
The board after step 5 — 1 of 12 queens placed. Step 6. The sea green region has 3 squares left, spread across rows 10 and 11, columns 9 and 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 7. Sea green and teal are now boxed into rows 10 and 11 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 10 squares at once.
Step 8. Here is the pairing argument. The amber, blue, mint, and slate regions have no open squares outside rows 2–5 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 14 squares come off in one stroke.
Step 9. The pale yellow region has 3 squares left, spread across rows 6 and 7, columns 8 and 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 10. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 5. 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 11. Test row 1, column 6 for indigo: place a queen there, run the forced consequences through, and row 10 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 11 — 1 of 12 queens placed. Step 12. The indigo region is down to one open square, at row 1, column 4. Every colour holds exactly one queen, so that square is forced. Row 1 and column 4 are now spoken for, along with the squares touching that queen.
Step 13. Wherever mint 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 14. 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: 6 squares come off.
Step 15. Column 11 has only khaki squares left (4 open squares). The column needs a queen and only khaki can provide one, which pins that colour to this column and kills its squares elsewhere: 3 squares.
Step 16. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: column 12 is down to one square, putting a queen on row 6, column 12; pale yellow has one square left, so its queen takes row 7, column 8; amber has one square left, so its queen takes row 2, column 7; row 5 is down to one square, putting a queen on row 5, column 1; row 4 is down to one square, putting a queen on row 4, column 6; teal has one square left, so its queen takes row 11, column 5; sea green has one square left, so its queen takes row 10, column 10; blue has one square left, so its queen takes row 3, column 9; khaki has one square left, so its queen takes row 8, column 11; orange has one square left, so its queen takes row 9, column 2. 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 9, column 2.
The board after step 16 — 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 #98
Board #98 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.
As with every puzzle in this collection, #98 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 #98 are: rose (1 square), pale yellow (4 squares), indigo (6 squares), amber (7 squares), sea green (7 squares), blue (9 squares), mint (10 squares), teal (10 squares), khaki (12 squares), slate (15 squares), red (24 squares), orange (39 squares). An even split would hand every color 12 squares, so the rose region is running 11 squares under that average while the orange region is the roomiest area on the grid with 39 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Rose has only 1 place to put its queen; orange has 39. The spread between them on this board is 38 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Look at arrangement as well as area. Rose is the densest region on this grid at 100% of a 1×1 frame; red is the loosest, occupying only 44% of its 9×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Below is every color on #98 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.
Rose occupies 1 square within rows 12–12, columns 3–3. It never leaves row 12, so its queen is locked to that row.
Pale yellow: 4 squares across rows 5–7, columns 8–9.
Indigo covers 6 squares, spanning 2 rows and 5 columns (rows 1–2, columns 3–7).
Amber occupies 7 squares within rows 1–3, columns 7–11.
Sea green: 7 squares across rows 10–12, columns 9–12.
Blue covers 9 squares, spanning 4 rows and 5 columns (rows 1–4, columns 8–12).
Mint occupies 10 squares within rows 2–6, columns 4–7.
Teal: 10 squares across rows 10–12, columns 4–8.
Khaki covers 12 squares, spanning 7 rows and 2 columns (rows 5–11, columns 11–12).
Slate occupies 15 squares within rows 1–5, columns 1–5.
Red: 24 squares across rows 3–11, columns 7–12.
Orange covers 39 squares, spanning 8 rows and 7 columns (rows 5–12, columns 1–7).
Where to start on this board
Rose is the natural opening on #98: it is the smallest region on the grid at 1 square, boxed into rows 12–12 and columns 3–3. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of 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. Indigo 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 #98
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 8 and 9 are the narrowest, touched by only 3 colors. Columns 1 and 2 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 5, meeting 6 colors, is the opposite case — the busiest line on #98 and the last one you should expect to crack.
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.
Keep alternating between two views of the board. In the region view you ask which of the twelve colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 12x12 board unwinds.
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 #98 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 38. 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.
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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #98 the place to look for one is around column 1.
Here is a shape check that gives nothing away: when #98 is solved, 4 queens sit on the outer edge of the grid and 8 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #98?
Puzzle #98 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #98 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #98, 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 #98?
Start with the rose region — the most constrained color on this board, with 1 square spanning rows 12–12 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.