11x11 Queens Puzzle #15
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 #15
What follows is a complete solve of puzzle #15, 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 rows.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, last square in a column.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Only row 5, column 2 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 2, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. The teal region is down to one open square, at row 11, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 11 and column 7, the remaining teal squares, and the ring of neighbours around row 11, column 7.
Step 3. Indigo cannot leave row 1 — every one of its open squares (3 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 6 squares in all.
Step 4. All of the orange region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
The board after step 4 — 2 of 11 queens placed. Step 5. Sea green is down to 4 candidates in rows 9 and 10, columns 9–11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.
Step 6. Here is the pairing argument. The orange and sea green regions have no open squares outside rows 9 and 10 — 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: 9 squares come off in one stroke.
Step 7. Only row 8, column 6 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 6, in khaki and in the ring around that square is now dead.
Step 8. The pale yellow region is down to one open square, at row 7, column 8. Every colour holds exactly one queen, so that square is forced. Row 7 and column 8 are now spoken for, along with the squares touching that queen.
The board after step 8 — 4 of 11 queens placed. Step 9. Only row 1, column 9 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 9, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 10. Wherever mint ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Red is down to 2 candidates in rows 3 and 6, columns 3 and 4, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about red itself.
Step 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: column 4 is down to one square, putting a queen on row 6, column 4; column 3 is down to one square, putting a queen on row 2, column 3; column 5 is down to one square, putting a queen on row 4, 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; orange has one square left, so its queen takes row 9, column 1. 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 1.
The board after step 12 — 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 #15
11x11 puzzle #15 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.
Like every board on this site, #15 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 #15 are: slate (1 square), teal (1 square), indigo (5 squares), orange (6 squares), pale yellow (8 squares), sea green (9 squares), khaki (10 squares), blue (11 squares), amber (16 squares), red (25 squares), mint (29 squares). An even split would hand every color 11 squares, so the slate region is running 10 squares under that average while the mint region is the roomiest area on the grid with 29 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 1 square while mint has 29 to spare, a gap of 28. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Slate is the densest region on this grid at 100% of a 1×1 frame; red is the loosest, occupying only 46% of its 9×6 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.
Slate: 1 square across rows 5–5, columns 2–2. It never leaves row 5, so its queen is locked to that row.
Teal covers 1 square, spanning 1 row and 1 column (rows 11–11, columns 7–7). It never leaves row 11, so its queen is locked to that row.
Indigo occupies 5 squares within rows 1–2, columns 6–9.
Orange: 6 squares across rows 9–11, columns 1–2.
Pale yellow covers 8 squares, spanning 3 rows and 5 columns (rows 7–9, columns 7–11).
Sea green occupies 9 squares within rows 9–11, columns 8–11.
Khaki: 10 squares across rows 8–10, columns 4–8.
Blue covers 11 squares, spanning 7 rows and 2 columns (rows 1–7, columns 10–11).
Amber occupies 16 squares within rows 1–5, columns 1–5.
Red: 25 squares across rows 3–11, columns 1–6.
Mint covers 29 squares, spanning 6 rows and 7 columns (rows 2–7, columns 4–10).
Where to start on this board
The most constrained color is where a solve should begin, and on #15 that is slate with 1 square. It reaches from row 5 to row 5 and from column 2 to column 2 — 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. Teal and indigo 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 #15
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6 and 8 are the narrowest, touched by only 3 colors. Column 3 is the tightest column, with 2 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 9 sits at the far end with 5 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.
Resist the plausible-looking queen. On #15 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 #15 is rated easy
The easy label comes out of four measurable things about #15: four of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 3 colors, and its largest and smallest regions differ by 28. Boards that score tight on those measures give up their first queen fast.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
If #15 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.
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 3 on this board.
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 #15?
Puzzle #15 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #15 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #15, 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 #15?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.