8x8 Queens Puzzle #121
Difficulty: hard. Place 8 queens on this 8x8 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 16-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 16 deductions — 8 queen placements and 8 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 line confined to one region.
Techniques this board needs: last square in a region, region shadow, region confined to one line, line confined to one region, last square in a column.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Only row 8, column 8 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 8, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 2. Indigo is down to 4 candidates in rows 1–3, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. Slate cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 6 squares in all.
Step 4. The indigo region is down to one open square, at row 1, column 6. Every colour holds exactly one queen, so that square is forced. Row 1 and column 6 are now spoken for, along with the squares touching that queen.
The board after step 4 — 2 of 8 queens placed. Step 5. Pale yellow cannot leave row 7 — every one of its open squares (2 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 5 squares in all.
Step 6. Every open square left in row 6 (3 open squares) belongs to the red region. Since that row must contain a queen, red is the colour that supplies it — so red squares anywhere else on the board are out, 3 squares in all.
Step 7. Scan column 5 and only row 6, column 5 remains — a red square. Every column takes a queen, so it takes this one. Cross out the whole of row 6 and column 5, the remaining red squares, and the ring of neighbours around row 6, column 5.
Step 8. Mint is down to 6 candidates in rows 2–5, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
The board after step 8 — 3 of 8 queens placed. Step 9. The amber region has 3 squares left, spread across rows 2 and 3, columns 2 and 3. 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. Wherever blue 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 11. Column 4 has only mint squares left (3 open squares). The column needs a queen and only mint can provide one, which pins that colour to this column and kills its squares elsewhere: 2 squares.
Step 12. 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: column 3 is down to one square, putting a queen on row 2, column 3; mint has one square left, so its queen takes row 4, column 4; slate has one square left, so its queen takes row 5, column 7; blue has one square left, so its queen takes row 3, column 1; pale yellow has one square left, so its queen takes row 7, 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 7, column 2.
The board after step 12 — 8 of 8 queens placed.
That completes the board: 8 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 8x8 Queens puzzle #121
Puzzle #121 is one of the hard-rated boards in the 8x8 set. Eight colors divide its 64 squares, and you win by placing eight 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 eight regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #121 are: khaki (1 square), pale yellow (5 squares), indigo (7 squares), blue (7 squares), mint (7 squares), slate (8 squares), amber (11 squares), red (18 squares). An even split would hand every color 8 squares, so the khaki region is running 7 squares under that average while the red region is the roomiest area on the grid with 18 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 khaki, with 1 square, is under far more pressure than red with 18. A 17-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Khaki is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 58% of its 4×3 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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.
Khaki covers 1 square, spanning 1 row and 1 column (rows 8–8, columns 8–8). It never leaves row 8, so its queen is locked to that row.
Pale yellow occupies 5 squares within rows 7–8, columns 1–3.
Indigo: 7 squares across rows 1–3, columns 6–8.
Blue covers 7 squares, spanning 5 rows and 2 columns (rows 2–6, columns 1–2).
Mint occupies 7 squares within rows 2–5, columns 2–4.
Slate: 8 squares across rows 4–8, columns 7–8.
Amber covers 11 squares, spanning 3 rows and 6 columns (rows 1–3, columns 1–6).
Red occupies 18 squares within rows 3–8, columns 3–7.
Where to start on this board
Open on the tightest color. Here that is khaki, confined to 1 square between rows 8 and 8 and columns 8 and 8. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #121.
Because khaki never leaves row 8, its queen must be in that row — and that instantly settles row 8 for everyone else. Sweep along it and X out every square that is not khaki. 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. Pale yellow 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 #121
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Column 5 is the tightest column, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 7 rows. Row 3, 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 64 squares of this grid. First: has any of the eight regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 8x8 boards almost always run one of those checks and forget the other.
Resist the plausible-looking queen. On #121 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #121 is rated hard
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 2 colors, and the gap between its largest and smallest regions is 17 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 hard board from the rest.
Stuck on #121 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 8x8
At 8x8 the whole grid stays within a single glance, which makes systematic scanning practical. Read the rows in order and count the colors in each — a row touched by two colors is far more constrained than one touched by five, and that is usually where the next forced move hides. The board is small enough that re-scanning from the top after each placement costs almost nothing.
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 row 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 5 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 8x8 Queens puzzle #121?
Puzzle #121 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 8x8 puzzle #121 have only one solution?
Yes. Every puzzle in the 8x8 collection, including #121, has been verified to have exactly one valid arrangement of eight 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 khaki region — the most constrained color on this board, with 1 square spanning rows 8–8 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.