11x11 Queens Puzzle #157
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 27-step reasoning is written out below.
Previous puzzle · All 11x11 puzzles · Next puzzle
Step-by-step solution for puzzle #157
What follows is a complete solve of puzzle #157, worked the way a person works it rather than the way a computer does. It runs to 27 deductions — 11 queen placements and 16 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 confined to one line, pairing argument on rows, pairing argument on columns, region shadow, proof by contradiction, last square in a column.
Show the full walkthrough (20 steps) — contains the solution
Step 1. Only row 10, column 1 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 1, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 2. All of the indigo region's remaining squares (3 open squares) lie in row 1. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 3. Orange and teal are now boxed into rows 9 and 11 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 10 squares at once.
Step 4. Here is the pairing argument. The blue, slate, 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: 12 squares come off in one stroke.
Step 5. Khaki is down to 3 candidates in rows 6 and 8, columns 7 and 8, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about khaki itself.
The board after step 5 — 1 of 11 queens placed. Step 6. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 6. Follow the forced moves that would create and the blue region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 7. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 1 square in all.
Step 8. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 3. Follow the forced moves that would create and column 9 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 9. Test row 3, column 4 for amber: place a queen there, run the forced consequences through, and column 9 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 10. Wherever amber 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.
The board after step 10 — 1 of 11 queens placed. Step 11. Test row 1, column 5 for indigo: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 12. The indigo region is down to one open square, at row 1, column 4. Every colour holds exactly one queen, so that square is forced. Row 1 and column 4 are now spoken for, along with the squares touching that queen.
Step 13. Test row 4, column 5 for amber: place a queen there, run the forced consequences through, and row 7 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. Amber is down to 2 candidates in rows 2 and 4, columns 2 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 15. Test row 4, column 6 for amber: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 15 — 2 of 11 queens placed. Step 16. The amber region is down to one open square, at row 2, column 2. Every colour holds exactly one queen, so that square is forced. Row 2 and column 2 are now spoken for, along with the squares touching that queen.
Step 17. 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: 6 squares in all.
Step 18. All of the slate region's remaining squares (3 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
Step 19. Mint and khaki 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 3 squares at once.
Step 20. 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: 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 11; blue has one square left, so its queen takes row 3, column 10; slate has one square left, so its queen takes row 5, column 9; column 8 is down to one square, putting a queen on row 8, column 8; red has one square left, so its queen takes row 6, column 5; mint has one square left, so its queen takes row 4, column 7; pale yellow has one square left, so its queen takes row 7, column 3. 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 7, column 3.
The board after step 20 — 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 #157
Puzzle #157 is one of the medium-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.
Like every board on this site, #157 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 #157 are: sea green (1 square), indigo (3 squares), slate (5 squares), blue (7 squares), orange (7 squares), teal (8 squares), amber (15 squares), khaki (15 squares), pale yellow (18 squares), mint (20 squares), red (22 squares). An even split would hand every color 11 squares, so the sea green region is running 10 squares under that average while the red region is the roomiest area on the grid with 22 squares.
Every color takes one queen and one only, whatever its size — which is why the 21 square difference between sea green and red 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.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×3 box that encloses it, while amber is the most scattered at 50% of its 5×6 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
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 medium board.
Sea green covers 1 square, spanning 1 row and 1 column (rows 10–10, columns 1–1). It never leaves row 10, so its queen is locked to that row.
Indigo occupies 3 squares within rows 1–1, columns 4–6. It never leaves row 1, so its queen is locked to that row.
Slate: 5 squares across rows 4–5, columns 9–11.
Blue covers 7 squares, spanning 3 rows and 3 columns (rows 1–3, columns 9–11).
Orange occupies 7 squares within rows 9–11, columns 9–11.
Teal: 8 squares across rows 9–11, columns 5–8.
Amber covers 15 squares, spanning 5 rows and 6 columns (rows 1–5, columns 1–6).
Khaki occupies 15 squares within rows 6–9, columns 7–11.
Pale yellow: 18 squares across rows 3–9, columns 1–5.
Mint covers 20 squares, spanning 7 rows and 5 columns (rows 1–7, columns 5–9).
Red occupies 22 squares within rows 5–11, columns 1–6.
Where to start on this board
Sea green is the natural opening on #157: it is the smallest region on the grid at 1 square, boxed into rows 10–10 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 sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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 and slate 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 #157
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 8 and 11 are the narrowest, touched by only 3 colors. Columns 2, 3, 7 and 8 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 10 rows. Row 9, meeting 5 colors, is the opposite case — the busiest line on #157 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.
The whole middle game reduces to two sweeps. Sweep the eleven colors looking for one with a single square left; sweep the eleven rows and eleven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 11x11 solve from a stalled one.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #157 is rated medium
The medium label comes out of four measurable things about #157: three 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 21. Boards that score tight on those measures give up their first queen fast.
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 medium board from the rest.
If #157 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.
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 #157 is row 2.
Here is a shape check that gives nothing away: when #157 is solved, 4 queens sit on the outer edge of the grid and 7 sit inside it, and no queen lands 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
- 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.
Full rules · Strategy library · Puzzle statistics
Questions about this puzzle
How hard is 11x11 Queens puzzle #157?
Puzzle #157 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #157 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #157, 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 #157?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.