9x9 Queens Puzzle #134
Difficulty: hard. Place 9 queens on this 9x9 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 15-step reasoning is written out below.
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Step-by-step solution for puzzle #134
What follows is a complete solve of puzzle #134, worked the way a person works it rather than the way a computer does. It runs to 15 deductions — 9 queen placements and 6 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 region shadow.
Techniques this board needs: last square in a region, region shadow, region confined to one line, last square in a row.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 9, column 2 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 2, the rest of the orange region and every square adjacent to it — diagonals included.
Step 2. The pale yellow region is down to one open square, at row 7, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 1, the remaining pale yellow squares, and the ring of neighbours around row 7, column 1.
Step 3. Only row 8, column 9 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 9, in khaki and in the ring around that square is now dead.
Step 4. Wherever indigo ends up among its 3 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 — 3 of 9 queens placed. Step 5. Blue is down to 7 candidates in rows 1–4, columns 3–6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 6. The mint region has 6 squares left, spread across rows 3–5, columns 6–8. 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 7. Slate cannot leave column 3 — every one of its open squares (3 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 3 squares in all.
The board after step 7 — 3 of 9 queens placed. Step 8. 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 9. Amber cannot leave row 2 — every one of its open squares (3 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 1 square in all.
Step 10. All of the blue region's remaining squares (2 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
Step 11. 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: row 4 is down to one square, putting a queen on row 4, column 8; amber has one square left, so its queen takes row 2, column 7; blue has one square left, so its queen takes row 3, column 5; 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 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 5, column 3.
The board after step 11 — 9 of 9 queens placed.
That completes the board: 9 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 9x9 Queens puzzle #134
Board #134 of the 9x9 bank carries a hard rating. Its 81 squares are shared between nine colored regions, and the solution places nine 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 #134, 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 nine regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #134 are: orange (1 square), khaki (4 squares), indigo (5 squares), pale yellow (7 squares), amber (8 squares), blue (8 squares), mint (10 squares), slate (14 squares), red (24 squares). An even split would hand every color 9 squares, so the orange region is running 8 squares under that average while the red region is the roomiest area on the grid with 24 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Orange has only 1 place to put its queen; red has 24. The spread between them on this board is 23 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here amber is the most solid, filling 100% of its 2×4 box, and blue the most diffuse at 50% of its 4×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Orange occupies 1 square within rows 9–9, columns 2–2. It never leaves row 9, so its queen is locked to that row.
Khaki: 4 squares across rows 7–9, columns 8–9.
Indigo covers 5 squares, spanning 2 rows and 4 columns (rows 1–2, columns 1–4).
Pale yellow occupies 7 squares within rows 7–9, columns 1–3.
Amber: 8 squares across rows 1–2, columns 6–9.
Blue covers 8 squares, spanning 4 rows and 4 columns (rows 1–4, columns 3–6).
Mint occupies 10 squares within rows 3–6, columns 6–9.
Slate: 14 squares across rows 2–7, columns 1–3.
Red covers 24 squares, spanning 6 rows and 5 columns (rows 4–9, columns 4–8).
Where to start on this board
Open on the tightest color. Here that is orange, confined to 1 square between rows 9 and 9 and columns 2 and 2. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #134.
Because orange never leaves row 9, its queen must be in that row — and that instantly settles row 9 for everyone else. Sweep along it and X out every square that is not orange. 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 and amber are no more than two rows tall, so their queens are 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 #134
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 5, 6 and 8 are the narrowest, touched by only 3 colors. Column 5 is the tightest column, with 2 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 8 rows. Row 9 sits at the far end with 4 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #134 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
Keep alternating between two views of the board. In the region view you ask which of the nine 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 9x9 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 #134 is rated hard
The hard label comes out of four measurable things about #134: two of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 23. 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 #134 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 9x9
At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.
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 #134 the place to look for one is around column 5.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 5 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
- 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 9x9 Queens puzzle #134?
Puzzle #134 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 9x9 puzzle #134 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #134, has been verified to have exactly one valid arrangement of nine queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #134?
Start with the orange region — the most constrained color on this board, with 1 square spanning rows 9–9 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.