9x9 Queens Puzzle #129
Difficulty: medium. 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 21-step reasoning is written out below.
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Step-by-step solution for puzzle #129
What follows is a complete solve of puzzle #129, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 9 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, line confined to one region, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (18 steps) — contains the solution
Step 1. Only row 1, column 9 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 9, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The mint region is down to one open square, at row 4, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 6, the remaining mint squares, and the ring of neighbours around row 4, column 6.
Step 3. Blue cannot leave column 4 — 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: 8 squares in all.
Step 4. Wherever slate 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.
Step 5. Orange is down to 4 candidates in rows 7–9, columns 2 and 3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
The board after step 5 — 2 of 9 queens placed. Step 6. Here is the pairing argument. The amber and blue regions have no open squares outside rows 2 and 3 — 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: 4 squares come off in one stroke.
Step 7. Column 1 has only pale yellow squares left (5 open squares). The column needs a queen and only pale yellow can provide one, which pins that colour to this column and kills its squares elsewhere: 2 squares.
Step 8. Here is the pairing argument. The amber and slate regions have no open squares outside columns 7 and 8 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 5 squares come off in one stroke.
Step 9. The khaki region has 3 squares left, spread across rows 7–9, columns 3 and 5. 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.
The board after step 9 — 2 of 9 queens placed. Step 10. All of the red region's remaining squares (3 open squares) lie in row 6. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.
Step 11. Only row 5, column 8 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 8, in slate and in the ring around that square is now dead.
Step 12. The amber region is down to one open square, at row 2, column 7. Every colour holds exactly one queen, so that square is forced. Row 2 and column 7 are now spoken for, along with the squares touching that queen.
Step 13. Only row 3, column 4 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 4, the rest of the blue region and every square adjacent to it — diagonals included.
Step 14. All of the pale yellow region's remaining squares (3 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
The board after step 14 — 5 of 9 queens placed. Step 15. Test row 6, column 2 for red: place a queen there, run the forced consequences through, and row 7 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. Every open square left in column 2 (2 open squares) belongs to the orange region. Since that column must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 1 square in all.
Step 17. 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: 1 square in all.
Step 18. 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: row 8 is down to one square, putting a queen on row 8, column 5; red has one square left, so its queen takes row 6, column 3; orange has one square left, so its queen takes row 9, column 2; pale yellow has one square left, so its queen takes row 7, 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 7, column 1.
The board after step 18 — 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 #129
Puzzle #129 is one of the medium-rated boards in the 9x9 set. Nine colors divide its 81 squares, and you win by placing nine queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
There is exactly one way to finish #129, 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 #129 are: indigo (1 square), mint (1 square), orange (4 squares), blue (5 squares), pale yellow (9 squares), slate (10 squares), amber (13 squares), khaki (14 squares), red (24 squares). An even split would hand every color 9 squares, so the indigo region is running 8 squares under that average while the red region is the roomiest area on the grid with 24 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So indigo, with 1 square, is under far more pressure than red with 24. A 23-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 43% of its 7×3 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 medium board.
Indigo: 1 square across rows 1–1, columns 9–9. It never leaves row 1, so its queen is locked to that row.
Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 6–6). It never leaves row 4, so its queen is locked to that row.
Orange occupies 4 squares within rows 7–9, columns 2–3.
Blue: 5 squares across rows 2–4, columns 4–6.
Pale yellow covers 9 squares, spanning 7 rows and 3 columns (rows 3–9, columns 1–3).
Slate occupies 10 squares within rows 4–7, columns 6–9.
Amber: 13 squares across rows 1–4, columns 5–9.
Khaki covers 14 squares, spanning 3 rows and 7 columns (rows 7–9, columns 3–9).
Red occupies 24 squares within rows 1–7, columns 1–8.
Where to start on this board
The most constrained color is where a solve should begin, and on #129 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 9 to column 9 — 1 row 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 indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Mint 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 #129
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 5, 6, 8 and 9 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 8 rows. Row 4 is the crowded extreme at 6 colors — worth revisiting later, not now.
Working through the middle game
The middle of #129 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.
Two checks, run alternately, will carry you through the 81 squares of this grid. First: has any of the nine regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 9x9 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 #129 is rated medium
The medium label comes out of four measurable things about #129: four 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.
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 #129 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 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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #129, start hunting for it near column 1.
A non-spoiling distribution check: solved, #129 carries 3 queens on the perimeter against 6 queens inside, 1 of them cornered. 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 9x9 Queens puzzle #129?
Puzzle #129 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #129 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #129, 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 #129?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.