11x11 Queens Puzzle #135
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 21-step reasoning is written out below.
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Step-by-step solution for puzzle #135
What follows is a complete solve of puzzle #135, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 11 queen placements and 10 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 proof by contradiction.
Techniques this board needs: region confined to one line, region shadow, pairing argument on rows, last square in a region, pairing argument on columns, proof by contradiction, last square in a row.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Blue cannot leave row 3 — every one of its open squares (3 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 2. Slate is down to 3 candidates in rows 5 and 6, columns 5 and 6, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 3. The red region has 6 squares left, spread across rows 5–7, columns 1–3. 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.
Step 4. Wherever sea green ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 0 of 11 queens placed. Step 5. Orange is down to 4 candidates in rows 9–11, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
Step 6. Here is the pairing argument. The orange, sea green, and teal regions have no open squares outside rows 9–11 — 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: 11 squares come off in one stroke.
Step 7. Slate, red, pale yellow, and khaki are now boxed into rows 5–8 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 19 squares at once.
Step 8. The mint region is down to one open square, at row 4, column 11. Every colour holds exactly one queen, so that square is forced. Row 4 and column 11 are now spoken for, along with the squares touching that queen.
Step 9. Only row 3, column 9 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 9, the rest of the blue region and every square adjacent to it — diagonals included.
The board after step 9 — 2 of 11 queens placed. Step 10. 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: 7 squares come off.
Step 11. Red, khaki, and sea green are now boxed into columns 1–3 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 4 squares at once.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 8. Follow the forced moves that would create and column 10 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 13. 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: indigo has one square left, so its queen takes row 1, column 10; row 9 is down to one square, putting a queen on row 9, column 5; slate has one square left, so its queen takes row 5, column 6; row 10 is down to one square, putting a queen on row 10, column 3; red has one square left, so its queen takes row 6, column 1; khaki has one square left, so its queen takes row 8, column 2; pale yellow has one square left, so its queen takes row 7, column 4; amber has one square left, so its queen takes row 2, column 7; teal has one square left, so its queen takes row 11, 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 11, column 8.
The board after step 13 — 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 #135
11x11 puzzle #135 is rated easy. 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.
There is exactly one way to finish #135, 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 #135 are: blue (3 squares), slate (3 squares), sea green (4 squares), mint (5 squares), orange (5 squares), red (6 squares), indigo (7 squares), khaki (10 squares), teal (12 squares), pale yellow (17 squares), amber (49 squares). An even split would hand every color 11 squares, so the blue region is running 8 squares under that average while the amber region is the roomiest area on the grid with 49 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Blue has only 3 places to put its queen; amber has 49. The spread between them on this board is 46 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 1×3 frame; pale yellow is the loosest, occupying only 47% of its 6×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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 easy board.
Blue: 3 squares across rows 3–3, columns 9–11. It never leaves row 3, so its queen is locked to that row.
Slate covers 3 squares, spanning 2 rows and 2 columns (rows 5–6, columns 5–6).
Sea green occupies 4 squares within rows 10–11, columns 1–3.
Mint: 5 squares across rows 4–6, columns 10–11.
Orange covers 5 squares, spanning 3 rows and 2 columns (rows 9–11, columns 4–5).
Red occupies 6 squares within rows 5–7, columns 1–3.
Indigo: 7 squares across rows 1–2, columns 8–11.
Khaki covers 10 squares, spanning 5 rows and 3 columns (rows 6–10, columns 1–3).
Teal occupies 12 squares within rows 9–11, columns 6–11.
Pale yellow: 17 squares across rows 5–10, columns 3–8.
Amber covers 49 squares, spanning 9 rows and 11 columns (rows 1–9, columns 1–11).
Where to start on this board
The most constrained color is where a solve should begin, and on #135 that is blue with 3 squares. It reaches from row 3 to row 3 and from column 9 to column 11 — 1 row by 3 columns. 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 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. Indigo, slate and sea green 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 #135
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3 and 4 are the narrowest, touched by only 2 colors. Columns 4 and 7 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 6 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
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 #135 is rated easy
The easy label comes out of four measurable things about #135: one of its regions is locked to one row or column, its smallest color spans 3 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 46. Boards that score tight on those measures give up their first queen fast.
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.
Stuck on #135 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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #135 the place to look for one is around row 1.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 7 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #135?
Puzzle #135 is rated easy. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #135 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #135, 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 #135?
Start with the blue region — the most constrained color on this board, with 3 squares spanning rows 3–3 and columns 9–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.