9x9 Queens Puzzle #121
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 13-step reasoning is written out below.
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Step-by-step solution for puzzle #121
What follows is a complete solve of puzzle #121, worked the way a person works it rather than the way a computer does. It runs to 13 deductions — 9 queen placements and 4 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.
Show the full walkthrough (7 steps) — contains the solution
Step 1. Only row 4, column 9 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 9, the rest of the mint region and every square adjacent to it — diagonals included.
Step 2. The slate region is down to one open square, at row 5, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 4, the remaining slate squares, and the ring of neighbours around row 5, column 4.
Step 3. The indigo region has 3 squares left, spread across rows 1 and 2, 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.
Step 4. Wherever blue ends up among its 6 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 — 2 of 9 queens placed. Step 5. Indigo is down to 2 candidates in rows 1 and 2, columns 3 and 5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about indigo itself.
Step 6. All of the khaki region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 7. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 5; blue has one square left, so its queen takes row 3, column 6; amber has one square left, so its queen takes row 2, column 8; orange has one square left, so its queen takes row 9, column 1; khaki has one square left, so its queen takes row 8, column 3; red has one square left, so its queen takes row 6, column 2; pale yellow has one square left, so its queen takes row 7, 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 7, column 7.
The board after step 7 — 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 #121
Puzzle #121 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 #121, 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 #121 are: mint (1 square), slate (3 squares), orange (3 squares), indigo (5 squares), khaki (5 squares), amber (6 squares), blue (18 squares), pale yellow (18 squares), red (22 squares). An even split would hand every color 9 squares, so the mint region is running 8 squares under that average while the red region is the roomiest area on the grid with 22 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 mint, with 1 square, is under far more pressure than red with 22. A 21-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 mint is the most solid, filling 100% of its 1×1 box, and blue the most diffuse at 40% of its 9×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #121, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 9–9). It never leaves row 4, so its queen is locked to that row.
Slate occupies 3 squares within rows 4–5, columns 3–4.
Orange: 3 squares across rows 8–9, columns 1–2.
Indigo covers 5 squares, spanning 3 rows and 3 columns (rows 1–3, columns 3–5).
Khaki occupies 5 squares within rows 7–9, columns 3–4.
Amber: 6 squares across rows 1–3, columns 7–9.
Blue covers 18 squares, spanning 9 rows and 5 columns (rows 1–9, columns 5–9).
Pale yellow occupies 18 squares within rows 5–9, columns 6–9.
Red: 22 squares across rows 1–8, columns 1–4.
Where to start on this board
The most constrained color is where a solve should begin, and on #121 that is mint with 1 square. It reaches from row 4 to row 4 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 mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. Slate and orange 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 #121
A quick way to find the next move is to count how many different colors touch each line. On this board, row 6 is the narrowest, touched by only 3 colors. Columns 1, 2, 5 and 6 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 8 rows. Row 8, meeting 5 colors, is the opposite case — the busiest line on #121 and the last one you should expect to crack.
Working through the middle game
The middle of #121 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 #121 is rated medium
The rating reflects how much the board concedes up front. Puzzle #121 has two regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 21 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a medium board from the rest.
Stuck on #121 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 #121, start hunting for it near column 1.
Here is a shape check that gives nothing away: when #121 is solved, 3 queens sit on the outer edge of the grid and 6 sit inside it, with 1 queen 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 9x9 Queens puzzle #121?
Puzzle #121 is rated medium. 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 #121 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #121, 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 #121?
Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.