11x11 Queens Puzzle #115
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 #115
What follows is a complete solve of puzzle #115, 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 proof by contradiction.
Techniques this board needs: last square in a region, region shadow, pairing argument on rows, proof by contradiction, line confined to one region, last square in a column.
Show the full walkthrough (18 steps) — contains the solution
Step 1. Only row 5, column 11 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 11, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. The mint region is down to one open square, at row 4, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 2, the remaining mint squares, and the ring of neighbours around row 4, column 2.
Step 3. Only row 6, column 1 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 1, in red and in the ring around that square is now dead.
Step 4. Wherever amber 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.
Step 5. Pale yellow is down to 5 candidates in rows 3, 7, 8, and 9, columns 9 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
The board after step 5 — 3 of 11 queens placed. Step 6. The sea green region has 4 squares left, spread across rows 9 and 10, columns 8–10. 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 7. Indigo, amber, and blue are now boxed into rows 1–3 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 1 square at once.
Step 8. Pale yellow is down to 3 candidates in rows 7 and 8, columns 9 and 10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. Test row 1, column 4 for indigo: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 9 — 3 of 11 queens placed. Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 5. Follow the forced moves that would create and column 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 11. Test row 2, column 3 for indigo: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 3, column 4. Follow the forced moves that would create and column 5 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 13. Test row 3, column 5 for indigo: place a queen there, run the forced consequences through, and column 4 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. Indigo is down to 2 candidates in rows 1 and 3, columns 3 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
The board after step 14 — 3 of 11 queens placed. Step 15. Test row 3, column 6 for indigo: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. The indigo region is down to one open square, at row 1, column 3. Every colour holds exactly one queen, so that square is forced. Row 1 and column 3 are now spoken for, along with the squares touching that queen.
Step 17. Column 4 has only orange squares left (5 open squares). The column needs a queen and only orange can provide one, which pins that colour to this column and kills its squares elsewhere: 5 squares.
Step 18. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: column 5 is down to one square, putting a queen on row 2, column 5; blue has one square left, so its queen takes row 3, column 9; pale yellow has one square left, so its queen takes row 7, column 10; sea green has one square left, so its queen takes row 10, column 8; column 7 is down to one square, putting a queen on row 8, column 7; teal has one square left, so its queen takes row 11, column 6; orange has one square left, so its queen takes row 9, 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 9, column 4.
The board after step 18 — 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 #115
11x11 puzzle #115 is rated medium. 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.
Like every board on this site, #115 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 #115 are: slate (1 square), red (3 squares), mint (5 squares), sea green (7 squares), teal (8 squares), khaki (12 squares), blue (13 squares), amber (14 squares), pale yellow (15 squares), indigo (17 squares), orange (26 squares). An even split would hand every color 11 squares, so the slate region is running 10 squares under that average while the orange region is the roomiest area on the grid with 26 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Slate has only 1 place to put its queen; orange has 26. The spread between them on this board is 25 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. Slate is the densest region on this grid at 100% of a 1×1 frame; sea green is the loosest, occupying only 44% of its 4×4 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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.
Slate covers 1 square, spanning 1 row and 1 column (rows 5–5, columns 11–11). It never leaves row 5, so its queen is locked to that row.
Red occupies 3 squares within rows 6–7, columns 1–2.
Mint: 5 squares across rows 4–5, columns 1–4.
Sea green covers 7 squares, spanning 4 rows and 4 columns (rows 8–11, columns 8–11).
Teal occupies 8 squares within rows 9–11, columns 6–10.
Khaki: 12 squares across rows 5–9, columns 6–8.
Blue covers 13 squares, spanning 4 rows and 6 columns (rows 1–4, columns 6–11).
Amber occupies 14 squares within rows 2–6, columns 4–8.
Pale yellow: 15 squares across rows 3–9, columns 9–11.
Indigo covers 17 squares, spanning 4 rows and 6 columns (rows 1–4, columns 1–6).
Orange occupies 26 squares within rows 6–11, columns 1–5.
Where to start on this board
The most constrained color is where a solve should begin, and on #115 that is slate with 1 square. It reaches from row 5 to row 5 and from column 11 to column 11 — 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 slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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. Mint 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 #115
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 3 and 5 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 10 rows. Row 9 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #115 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.
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 #115 is rated medium
Ratings track how generous the opening is. On #115 you get two regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 25-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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.
If #115 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.
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 #115 the place to look for one is around row 1.
A non-spoiling distribution check: solved, #115 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 #115?
Puzzle #115 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #115 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #115, 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 #115?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.