11x11 Queens Puzzle #138
Difficulty: medium. 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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #138
What follows is a complete solve of puzzle #138, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 11 queen placements and 13 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: region shadow, region confined to one line, pairing argument on rows, pairing argument on columns, last square in a region, line confined to one region, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Wherever indigo ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. All of the amber region's remaining squares (2 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 11 squares come off.
Step 3. The khaki region has 3 squares left, spread across rows 5 and 8, columns 9 and 10. 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.
Step 4. Wherever orange ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 4 — 0 of 11 queens placed. Step 5. Teal is down to 3 candidates in rows 10 and 11, columns 9 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about teal itself.
Step 6. Here is the pairing argument. The indigo and amber regions have no open squares outside rows 1 and 2 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 13 squares come off in one stroke.
Step 7. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 10 squares in all.
Step 8. Here is the pairing argument. The khaki and teal regions have no open squares outside columns 9 and 10 — 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: 7 squares come off in one stroke.
Step 9. Indigo and pale yellow are now boxed into columns 7 and 8 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 7 squares at once.
The board after step 9 — 0 of 11 queens placed. Step 10. The slate region is down to one open square, at row 5, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 6, the remaining slate squares, and the ring of neighbours around row 5, column 6.
Step 11. Only row 8, column 10 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 10, in khaki and in the ring around that square is now dead.
Step 12. The teal region is down to one open square, at row 11, column 9. Every colour holds exactly one queen, so that square is forced. Row 11 and column 9 are now spoken for, along with the squares touching that queen.
Step 13. Wherever orange 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.
The board after step 13 — 3 of 11 queens placed. Step 14. Sea green is down to 3 candidates in rows 9 and 10, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.
Step 15. The red region has 5 squares left, spread across rows 6, 7, 9, and 10, columns 3–5. 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 16. Every open square left in row 4 (2 open squares) belongs to the mint region. Since that row must contain a queen, mint is the colour that supplies it — so mint squares anywhere else on the board are out, 5 squares in all.
Step 17. 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: row 6 is down to one square, putting a queen on row 6, column 4; blue has one square left, so its queen takes row 3, column 3; mint has one square left, so its queen takes row 4, column 1; sea green has one square left, so its queen takes row 10, column 2; orange has one square left, so its queen takes row 9, column 5; pale yellow has one square left, so its queen takes row 7, column 8; indigo has one square left, so its queen takes row 1, column 7; amber has one square left, so its queen takes row 2, column 11. 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 2, column 11.
The board after step 17 — 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 #138
Board #138 of the 11x11 bank carries a medium rating. Its 121 squares are shared between eleven colored regions, and the solution places eleven queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Like every board on this site, #138 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #138 are: amber (2 squares), indigo (3 squares), orange (6 squares), teal (6 squares), khaki (8 squares), sea green (10 squares), blue (13 squares), mint (14 squares), pale yellow (18 squares), red (20 squares), slate (21 squares). An even split would hand every color 11 squares, so the amber region is running 9 squares under that average while the slate region is the roomiest area on the grid with 21 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 amber, with 2 squares, is under far more pressure than slate with 21. A 19-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Amber is the tidiest shape on #138, packing 100% of its 2×1 bounding box, whereas red straggles across its 7×6 box at just 48%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Amber: 2 squares across rows 1–2, columns 11–11. It sits wholly inside column 11, so its queen must be in that column.
Indigo covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 7–8).
Orange occupies 6 squares within rows 8–10, columns 4–6.
Teal: 6 squares across rows 9–11, columns 9–11.
Khaki covers 8 squares, spanning 5 rows and 3 columns (rows 4–8, columns 9–11).
Sea green occupies 10 squares within rows 9–11, columns 1–6.
Blue: 13 squares across rows 1–3, columns 1–6.
Mint covers 14 squares, spanning 6 rows and 3 columns (rows 3–8, columns 1–3).
Pale yellow occupies 18 squares within rows 6–11, columns 6–10.
Red: 20 squares across rows 4–10, columns 2–7.
Slate covers 21 squares, spanning 6 rows and 7 columns (rows 1–6, columns 5–11).
Where to start on this board
Begin where the choices are fewest. On this board that means amber, which owns just 2 squares inside a 2×1 window running from row 1 to row 2 and column 11 to column 11. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because amber stays inside column 11, its queen belongs to that column and no other region may use it. Cross out the rest of column 11 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Indigo 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 #138
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 11 are the narrowest, touched by only 3 colors. Columns 1 and 8 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 10 rows. Row 10 is the crowded extreme at 5 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.
With eleven queens to seat and eleven regions competing over 121 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 11x11 boards quickly and players who stall halfway.
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 #138 is rated medium
Ratings track how generous the opening is. On #138 you get one region pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 3 colors, and a 19-square gap between the largest and smallest regions. Those four numbers largely explain the medium 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.
If #138 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
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 #138, start hunting for it near row 3.
A non-spoiling distribution check: solved, #138 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 #138?
Puzzle #138 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #138 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #138, 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 #138?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.