11x11 Queens Puzzle #137

Difficulty: easy. 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 #137

What follows is a complete solve of puzzle #137, 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 pairing argument on columns.

Techniques this board needs: last square in a region, region shadow, region confined to one line, pairing argument on rows, pairing argument on columns, last square in a row.

Show the full walkthrough (9 steps) — contains the solution

  1. Step 1. Only row 3, column 1 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 1, the rest of the blue region and every square adjacent to it — diagonals included.

  2. Step 2. The teal region is down to one open square, at row 11, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 8, the remaining teal squares, and the ring of neighbours around row 11, column 8.

  3. Step 3. The mint region has 3 squares left, spread across rows 2 and 4, 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.

    Puzzle #137 after step 3: 2 of 11 queens placed
    The board after step 3 — 2 of 11 queens placed.

  4. Step 4. 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: 6 squares come off.

  5. Step 5. Red is down to 3 candidates in rows 6 and 7, columns 3 and 4, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about red itself.

  6. Step 6. The orange region has 3 squares left, spread across rows 9 and 10, 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.

    Puzzle #137 after step 6: 2 of 11 queens placed
    The board after step 6 — 2 of 11 queens placed.

  7. Step 7. Orange and sea green are now boxed into rows 9 and 10 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 4 squares at once.

  8. Step 8. Here is the pairing argument. The indigo, mint, and orange regions have no open squares outside columns 9–11 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 14 squares come off in one stroke.

  9. 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: khaki has one square left, so its queen takes row 8, column 7; pale yellow has one square left, so its queen takes row 7, column 2; red has one square left, so its queen takes row 6, column 4; row 5 is down to one square, putting a queen on row 5, column 6; row 9 is down to one square, putting a queen on row 9, column 9; indigo has one square left, so its queen takes row 1, column 11; mint has one square left, so its queen takes row 4, column 10; amber has one square left, so its queen takes row 2, column 3; sea green has one square left, so its queen takes row 10, 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 10, column 5.

    Puzzle #137 after step 9: 11 of 11 queens placed
    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 #137

Puzzle #137 is one of the easy-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 #137, 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 #137 are: blue (1 square), teal (1 square), red (3 squares), indigo (4 squares), mint (6 squares), orange (9 squares), pale yellow (14 squares), sea green (16 squares), khaki (18 squares), amber (23 squares), slate (26 squares). An even split would hand every color 11 squares, so the blue region is running 10 squares under that average while the slate region is the roomiest area on the grid with 26 squares.

Every color takes one queen and one only, whatever its size — which is why the 25 square difference between blue and slate is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #137, packing 100% of its 1×1 bounding box, whereas pale yellow straggles across its 4×6 box at just 58%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.

Region by region

Below is every color on #137 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Blue occupies 1 square within rows 3–3, columns 1–1. It never leaves row 3, so its queen is locked to that row.

Teal: 1 square across rows 11–11, columns 8–8. It never leaves row 11, so its queen is locked to that row.

Red covers 3 squares, spanning 2 rows and 2 columns (rows 6–7, columns 3–4).

Indigo occupies 4 squares within rows 1–2, columns 9–11.

Mint: 6 squares across rows 2–4, columns 9–11.

Orange covers 9 squares, spanning 3 rows and 4 columns (rows 9–11, columns 8–11).

Pale yellow occupies 14 squares within rows 6–9, columns 1–6.

Sea green: 16 squares across rows 9–11, columns 1–7.

Khaki covers 18 squares, spanning 6 rows and 5 columns (rows 4–9, columns 7–11).

Amber occupies 23 squares within rows 1–6, columns 1–5.

Slate: 26 squares across rows 1–7, columns 5–9.

Where to start on this board

Blue is the natural opening on #137: it is the smallest region on the grid at 1 square, boxed into rows 3–3 and columns 1–1. 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 blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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, indigo 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 #137

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 8 and 10 are the narrowest, touched by only 2 colors. Columns 2, 6 and 7 are the tightest columns, with 3 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 10 rows. Row 6, meeting 5 colors, is the opposite case — the busiest line on #137 and the last one you should expect to crack.

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.

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.

Resist the plausible-looking queen. On #137 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 #137 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 25. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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 easy board from the rest.

Stuck on #137 despite the easy 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.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #137 is row 8.

Here is a shape check that gives nothing away: when #137 is solved, 3 queens sit on the outer edge of the grid and 8 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

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. 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.
  5. 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 #137?

Puzzle #137 is rated easy. 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 #137 have only one solution?

Yes. Every puzzle in the 11x11 collection, including #137, 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 #137?

Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.