11x11 Queens Puzzle #126
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 32-step reasoning is written out below.
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Step-by-step solution for puzzle #126
What follows is a complete solve of puzzle #126, worked the way a person works it rather than the way a computer does. It runs to 32 deductions — 11 queen placements and 21 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 columns, proof by contradiction, line confined to one region, last square in a region, last square in a row.
Show the full walkthrough (23 steps) — contains the solution
Step 1. Pale yellow cannot leave column 11 — every one of its open squares (3 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 9 squares in all.
Step 2. All of the orange region's remaining squares (2 open squares) lie in column 10. 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 indigo region has 4 squares left, spread across rows 1–3, columns 8 and 9. 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. All of the sea green 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: 11 squares come off.
Step 5. Indigo and red are now boxed into columns 8 and 9 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 9 squares at once.
Step 6. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 9. Follow the forced moves that would create and row 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 6 — 0 of 11 queens placed. Step 7. Test row 3, column 9 for indigo: place a queen there, run the forced consequences through, and row 4 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 8. 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: 6 squares come off.
Step 9. Amber, blue, slate, and teal are now boxed into columns 4–7 and nothing else. That is 4 colours needing 4 columns, so they fill those columns between them — which locks every remaining colour out and removes 6 squares at once.
Step 10. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 6. Follow the forced moves that would create and row 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 11. Test row 2, column 7 for amber: place a queen there, run the forced consequences through, and row 4 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 4. Follow the forced moves that would create and row 4 runs out of open squares entirely — impossible. So that square is ruled out for good.
The board after step 12 — 0 of 11 queens placed. Step 13. Test row 3, column 5 for amber: place a queen there, run the forced consequences through, and row 4 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 4, column 4. Follow the forced moves that would create and row 5 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. The amber region has 4 squares left, spread across rows 2 and 4, columns 5–7. 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 column 4 (2 open squares) belongs to the slate region. Since that column must contain a queen, slate is the colour that supplies it — so slate squares anywhere else on the board are out, 7 squares in all.
Step 17. Slate cannot leave column 4 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 3 squares in all.
The board after step 17 — 0 of 11 queens placed. Step 18. Now a short proof by contradiction. Suppose the amber queen went to row 4, column 5. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Test row 4, column 6 for amber: place a queen there, run the forced consequences through, and row 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 20. Now a short proof by contradiction. Suppose the amber queen went to row 4, column 7. Follow the forced moves that would create and row 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 21. Only row 2, column 5 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 5, the rest of the amber region and every square adjacent to it — diagonals included.
Step 22. 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, 3 squares in all.
Step 23. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: row 3 is down to one square, putting a queen on row 3, column 7; teal has one square left, so its queen takes row 11, column 6; sea green has one square left, so its queen takes row 10, column 1; orange has one square left, so its queen takes row 9, column 10; pale yellow has one square left, so its queen takes row 7, column 11; row 5 is down to one square, putting a queen on row 5, column 4; mint has one square left, so its queen takes row 4, column 2; khaki has one square left, so its queen takes row 8, column 3; red has one square left, so its queen takes row 6, column 9; indigo has one square left, so its queen takes row 1, 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 1, column 8.
The board after step 23 — 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 #126
Board #126 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.
Puzzle #126 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #126 are: sea green (2 squares), pale yellow (3 squares), orange (5 squares), slate (9 squares), indigo (10 squares), blue (14 squares), mint (14 squares), red (14 squares), teal (14 squares), amber (15 squares), khaki (21 squares). An even split would hand every color 11 squares, so the sea green region is running 9 squares under that average while the khaki 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 sea green, with 2 squares, is under far more pressure than khaki 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. Pale yellow is the tidiest shape on #126, packing 100% of its 3×1 bounding box, whereas slate straggles across its 6×4 box at just 38%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Below is every color on #126 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Sea green: 2 squares across rows 10–11, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Pale yellow covers 3 squares, spanning 3 rows and 1 column (rows 7–9, columns 11–11). It sits wholly inside column 11, so its queen must be in that column.
Orange occupies 5 squares within rows 8–11, columns 10–11.
Slate: 9 squares across rows 5–10, columns 4–7.
Indigo covers 10 squares, spanning 3 rows and 4 columns (rows 1–3, columns 8–11).
Blue occupies 14 squares within rows 2–6, columns 5–9.
Mint: 14 squares across rows 1–6, columns 1–4.
Red covers 14 squares, spanning 6 rows and 4 columns (rows 4–9, columns 8–11).
Teal occupies 14 squares within rows 8–11, columns 5–10.
Amber: 15 squares across rows 1–4, columns 2–7.
Khaki covers 21 squares, spanning 7 rows and 4 columns (rows 5–11, columns 1–4).
Where to start on this board
Begin where the choices are fewest. On this board that means sea green, which owns just 2 squares inside a 2×1 window running from row 10 to row 11 and column 1 to column 1. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because sea green stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: orange spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #126
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 3 colors. Columns 1, 2 and 3 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 9 is the crowded extreme at 6 colors — worth revisiting later, not now.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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 #126 is rated medium
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 19. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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 #126, start hunting for it near row 1.
A non-spoiling distribution check: solved, #126 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 #126?
Puzzle #126 is rated medium. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #126 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #126, 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 #126?
Start with the sea green region — the most constrained color on this board, with 2 squares spanning rows 10–11 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.