11x11 Queens Puzzle #121
Difficulty: hard. Place 11 queens on this 11x11 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 17-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 17 deductions — 11 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.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Only row 4, column 3 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 3, the rest of the mint region and every square adjacent to it — diagonals included.
Step 2. The teal region is down to one open square, at row 11, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 6, the remaining teal squares, and the ring of neighbours around row 11, column 6.
Step 3. The amber region has 4 squares left, spread across rows 1 and 2, columns 1, 2, and 4. 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 3 — 2 of 11 queens placed. Step 4. 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 5. Slate cannot leave column 1 — 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 6. The red region has 4 squares left, spread across rows 5 and 6, columns 9–11. 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.
The board after step 6 — 2 of 11 queens placed. Step 7. Wherever blue 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 8. All of the pale yellow region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 4; blue has one square left, so its queen takes row 3, column 11; slate has one square left, so its queen takes row 5, column 1; red has one square left, so its queen takes row 6, column 10; pale yellow has one square left, so its queen takes row 7, column 2; khaki has one square left, so its queen takes row 8, column 9; sea green has one square left, so its queen takes row 10, column 8; indigo has one square left, so its queen takes row 1, column 7; orange has one square left, so its queen takes row 9, column 5. 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 5.
The board after step 9 — 11 of 11 queens placed.
That completes the board: 11 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 11x11 Queens puzzle #121
Puzzle #121 is one of the hard-rated boards in the 11x11 set. Eleven colors divide its 121 squares, and you win by placing eleven 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 eleven 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), teal (1 square), red (4 squares), slate (6 squares), blue (7 squares), amber (9 squares), pale yellow (10 squares), sea green (11 squares), khaki (16 squares), indigo (24 squares), orange (32 squares). An even split would hand every color 11 squares, so the mint region is running 10 squares under that average while the orange region is the roomiest area on the grid with 32 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 orange with 32. A 31-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The mint region is the most compact color here, filling 100% of the 1×1 box that encloses it, while khaki is the most scattered at 53% of its 6×5 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
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 3–3). It never leaves row 4, so its queen is locked to that row.
Teal occupies 1 square within rows 11–11, columns 6–6. It never leaves row 11, so its queen is locked to that row.
Red: 4 squares across rows 5–6, columns 9–11.
Slate covers 6 squares, spanning 4 rows and 2 columns (rows 2–5, columns 1–2).
Blue occupies 7 squares within rows 2–5, columns 9–11.
Amber: 9 squares across rows 1–3, columns 1–4.
Pale yellow covers 10 squares, spanning 5 rows and 3 columns (rows 5–9, columns 1–3).
Sea green occupies 11 squares within rows 7–11, columns 6–9.
Khaki: 16 squares across rows 6–11, columns 7–11.
Indigo covers 24 squares, spanning 5 rows and 8 columns (rows 1–5, columns 4–11).
Orange occupies 32 squares within rows 4–11, columns 1–7.
Where to start on this board
Mint is the natural opening on #121: it is the smallest region on the grid at 1 square, boxed into rows 4–4 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 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. Teal and red 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 1 is the narrowest, touched by only 2 colors. Column 5 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 10 rows. Row 5 is the crowded extreme at 6 colors — worth revisiting later, not now.
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.
The whole middle game reduces to two sweeps. Sweep the eleven colors looking for one with a single square left; sweep the eleven rows and eleven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 11x11 solve from a stalled one.
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 hard
The rating reflects how much the board concedes up front. Puzzle #121 has four 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 31 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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 #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 11x11
At 11x11 you are managing 121 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.
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.
A non-spoiling distribution check: solved, #121 carries 4 queens on the perimeter against 7 queens inside, with the corners empty. 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 11x11 Queens puzzle #121?
Puzzle #121 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #121 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #121, has been verified to have exactly one valid arrangement of eleven 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 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.