11x11 Queens Puzzle #276
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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #276
What follows is a complete solve of puzzle #276, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 11 queen placements and 7 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 pairing argument on rows.
Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on rows, line confined to one region, last square in a column.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 10, column 3 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 3, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 2. Amber is down to 4 candidates in rows 1 and 2, columns 9–11, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
Step 3. Orange cannot leave column 1 — 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: 8 squares in all.
Step 4. All of the pale yellow region's remaining squares (3 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.
The board after step 4 — 1 of 11 queens placed. Step 5. Indigo and amber are now boxed into rows 1 and 2 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 5 squares at once.
Step 6. The blue region is down to one open square, at row 3, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 5, the remaining blue squares, and the ring of neighbours around row 3, column 5.
Step 7. Mint cannot leave row 4 — every one of its open squares (2 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 4 squares in all.
The board after step 7 — 2 of 11 queens placed. Step 8. The slate region is down to one open square, at row 5, column 7. Every colour holds exactly one queen, so that square is forced. Row 5 and column 7 are now spoken for, along with the squares touching that queen.
Step 9. The indigo region has 2 squares left, spread across rows 1 and 2, columns 6 and 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 10. Every open square left in column 4 (3 open squares) belongs to the khaki region. Since that column must contain a queen, khaki is the colour that supplies it — so khaki squares anywhere else on the board are out, 7 squares in all.
Step 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: column 6 is down to one square, putting a queen on row 1, column 6; amber has one square left, so its queen takes row 2, column 9; mint has one square left, so its queen takes row 4, column 11; red has one square left, so its queen takes row 6, column 10; teal has one square left, so its queen takes row 11, column 8; orange has one square left, so its queen takes row 9, column 1; 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 4. 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 8, column 4.
The board after step 11 — 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 #276
This is puzzle number 276 in the 11x11 collection, rated hard. The grid holds 121 squares split between eleven colored regions, and a finished board carries eleven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #276, 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 #276 are: sea green (1 square), amber (4 squares), indigo (6 squares), orange (6 squares), pale yellow (7 squares), slate (8 squares), mint (10 squares), teal (14 squares), red (16 squares), blue (20 squares), khaki (29 squares). An even split would hand every color 11 squares, so the sea green region is running 10 squares under that average while the khaki region is the roomiest area on the grid with 29 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Sea green is working with 1 square while khaki has 29 to spare, a gap of 28. Attack the cramped end and the roomy regions tend to resolve themselves.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 2×3 box, and red the most diffuse at 53% of its 6×5 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.
Sea green: 1 square across rows 10–10, columns 3–3. It never leaves row 10, so its queen is locked to that row.
Amber covers 4 squares, spanning 2 rows and 3 columns (rows 1–2, columns 9–11).
Indigo occupies 6 squares within rows 1–2, columns 6–8.
Orange: 6 squares across rows 9–11, columns 1–2.
Pale yellow covers 7 squares, spanning 3 rows and 3 columns (rows 6–8, columns 1–3).
Slate occupies 8 squares within rows 3–5, columns 4–8.
Mint: 10 squares across rows 2–4, columns 6–11.
Teal covers 14 squares, spanning 5 rows and 5 columns (rows 7–11, columns 7–11).
Red occupies 16 squares within rows 4–9, columns 7–11.
Blue: 20 squares across rows 1–5, columns 1–5.
Khaki covers 29 squares, spanning 7 rows and 6 columns (rows 5–11, columns 3–8).
Where to start on this board
Open on the tightest color. Here that is sea green, confined to 1 square between rows 10 and 10 and columns 3 and 3. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #276.
Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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 #276
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 6 and 11 are the narrowest, touched by only 3 colors. Columns 1, 2, 4 and 5 are the tightest columns, with 3 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 10 rows. Row 10, meeting 4 colors, is the opposite case — the busiest line on #276 and the last one you should expect to crack.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
Two checks, run alternately, will carry you through the 121 squares of this grid. First: has any of the eleven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 11x11 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 #276 is rated hard
The hard label comes out of four measurable things about #276: 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 28. 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 #276 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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.
Here is a shape check that gives nothing away: when #276 is solved, 4 queens sit on the outer edge of the grid and 7 sit inside it, and no queen lands 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 11x11 Queens puzzle #276?
Puzzle #276 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 11x11 puzzle #276 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #276, 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 #276?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.