11x11 Queens Puzzle #151
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 23-step reasoning is written out below.
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Step-by-step solution for puzzle #151
What follows is a complete solve of puzzle #151, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 11 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 pairing argument on columns.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, line confined to one region, last square in a row.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 11, column 1 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 1, the rest of the teal region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (2 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
Step 3. The blue region has 3 squares left, spread across rows 2 and 3, columns 10 and 11. 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 slate ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 5. Pale yellow is down to 4 candidates in rows 6 and 7, columns 7–9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.
The board after step 5 — 1 of 11 queens placed. Step 6. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 7. Wherever orange ends up among its 5 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. Here is the pairing argument. The amber, blue, and mint regions have no open squares outside rows 2–4 — 3 colours confined to 3 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 7 squares come off in one stroke.
Step 9. The orange region has 3 squares left, spread across rows 7 and 9, 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.
The board after step 9 — 1 of 11 queens placed. Step 10. Here is the pairing argument. The indigo and orange regions have no open squares outside columns 2 and 3 — 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 11. Row 4 has only mint squares left (2 open squares). The row needs a queen and only mint can provide one, which pins that colour to this row and kills its squares elsewhere: 5 squares.
Step 12. Here is the pairing argument. The blue and sea green regions have no open squares outside columns 10 and 11 — 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 13. Mint and khaki are now boxed into columns 8 and 9 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 3 squares at once.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: pale yellow has one square left, so its queen takes row 7, column 7; slate has one square left, so its queen takes row 5, column 6; orange has one square left, so its queen takes row 9, column 2; indigo has one square left, so its queen takes row 1, column 3; row 6 is down to one square, putting a queen on row 6, column 4; amber has one square left, so its queen takes row 2, column 5; blue has one square left, so its queen takes row 3, column 11; sea green has one square left, so its queen takes row 10, column 10; khaki has one square left, so its queen takes row 8, column 9; mint has one square left, so its queen takes row 4, column 8. 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 4, column 8.
The board after step 14 — 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 #151
11x11 puzzle #151 is rated hard. Eleven colored regions cover its 121 squares, and your job is to seat eleven queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Puzzle #151 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #151 are: teal (1 square), indigo (2 squares), slate (3 squares), blue (4 squares), pale yellow (5 squares), sea green (8 squares), mint (13 squares), amber (15 squares), orange (16 squares), khaki (19 squares), red (35 squares). An even split would hand every color 11 squares, so the teal region is running 10 squares under that average while the red region is the roomiest area on the grid with 35 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Teal is working with 1 square while red has 35 to spare, a gap of 34. 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 1×2 box, and khaki the most diffuse at 53% of its 9×4 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Teal covers 1 square, spanning 1 row and 1 column (rows 11–11, columns 1–1). It never leaves row 11, so its queen is locked to that row.
Indigo occupies 2 squares within rows 1–1, columns 2–3. It never leaves row 1, so its queen is locked to that row.
Slate: 3 squares across rows 5–6, columns 6–7.
Blue covers 4 squares, spanning 3 rows and 2 columns (rows 1–3, columns 10–11).
Pale yellow occupies 5 squares within rows 5–7, columns 7–9.
Sea green: 8 squares across rows 8–11, columns 9–11.
Mint covers 13 squares, spanning 4 rows and 5 columns (rows 1–4, columns 6–10).
Amber occupies 15 squares within rows 1–4, columns 1–7.
Orange: 16 squares across rows 3–10, columns 1–3.
Khaki covers 19 squares, spanning 9 rows and 4 columns (rows 3–11, columns 8–11).
Red occupies 35 squares within rows 3–11, columns 2–7.
Where to start on this board
The most constrained color is where a solve should begin, and on #151 that is teal with 1 square. It reaches from row 11 to row 11 and from column 1 to column 1 — 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 teal never leaves row 11, its queen must be in that row — and that instantly settles row 11 for everyone else. Sweep along it and X out every square that is not teal. 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 slate 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 #151
A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 3 colors. Columns 4 and 5 are the tightest columns, with 2 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 3 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
The middle of #151 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.
Keep alternating between two views of the board. In the region view you ask which of the eleven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 11x11 board unwinds.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #151 is rated hard
Ratings track how generous the opening is. On #151 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 34-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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 column 4 on this board.
As a sanity check rather than a spoiler: the finished solution to #151 puts 3 queens on the outer border and 8 queens in the interior, 1 of them in an actual corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #151?
Puzzle #151 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #151 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #151, 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 #151?
Start with the teal region — the most constrained color on this board, with 1 square spanning rows 11–11 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.