11x11 Queens Puzzle #117

Difficulty: hard. 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 36-step reasoning is written out below.

Previous puzzle · All 11x11 puzzles · Next puzzle

Step-by-step solution for puzzle #117

What follows is a complete solve of puzzle #117, worked the way a person works it rather than the way a computer does. It runs to 36 deductions — 11 queen placements and 25 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, last square in a region, pairing argument on rows, last square in a column, last square in a row.

Show the full walkthrough (33 steps) — contains the solution

Step 1. Indigo cannot leave row 1 — every one of its open squares (4 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 7 squares in all.
Step 2. Slate is down to 3 candidates in rows 5 and 6, columns 10 and 11, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 3. The orange region has 7 squares left, spread across rows 7–9, columns 4–6. 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 5 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 5. Pale yellow and sea green are now boxed into columns 1 and 2 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 8 squares at once.
Step 6. All of the amber region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.
Step 7. 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: 1 square in all.
Step 8. Khaki is down to 3 candidates in rows 6 and 8, columns 4 and 5, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about khaki itself.
The board after step 8 — 0 of 11 queens placed.
Step 9. Indigo, khaki, and orange are now boxed into columns 4–6 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 17 squares at once.
Step 10. Wherever mint ends up among its 7 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Test row 1, column 4 for indigo: place a queen there, run the forced consequences through, and the khaki region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 6. 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. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 5, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 14. All of the khaki region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 15. Orange cannot leave column 6 — every one of its open squares (3 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 1 square in all.
Step 16. Wherever mint 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 17. Test row 5, column 3 for amber: place a queen there, run the forced consequences through, and the teal region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 17 — 1 of 11 queens placed.
Step 18. The amber region is down to one open square, at row 2, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 3, the remaining amber squares, and the ring of neighbours around row 2, column 3.
Step 19. Wherever blue 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 20. Now a short proof by contradiction. Suppose the blue queen went to row 3, column 7. 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 21. Blue and slate are now boxed into columns 10 and 11 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 10 squares at once.
Step 22. Wherever red ends up among its 5 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 23. Test row 3, column 11 for blue: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 24. All of the blue 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: 2 squares come off.
Step 25. 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 6 squares at once.
The board after step 25 — 2 of 11 queens placed.
Step 26. The orange region is down to one open square, at row 9, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 6, the remaining orange squares, and the ring of neighbours around row 9, column 6.
Step 27. Scan column 7 and only row 11, column 7 remains — a teal square. Every column takes a queen, so it takes this one. 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 28. The sea green region is down to one open square, at row 10, column 1. Every colour holds exactly one queen, so that square is forced. Row 10 and column 1 are now spoken for, along with the squares touching that queen.
Step 29. Mint cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 2 squares in all.
Step 30. Row 7 has a single open square left, at column 2 in the pale yellow region. Every row needs a queen, so this one is forced. Row 7 and column 2 are now spoken for, along with the squares touching that queen.
Step 31. Look along row 8: one square survives, row 8, column 4, which belongs to khaki. A row cannot be left empty, so the queen goes there. That clears row 8, column 4, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 32. Now a short proof by contradiction. Suppose the blue queen went to row 4, column 10. 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 33. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: blue has one square left, so its queen takes row 3, column 10; mint has one square left, so its queen takes row 4, column 8; red has one square left, so its queen takes row 6, column 9; slate has one square left, so its queen takes row 5, column 11. 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 5, column 11.
The board after step 33 — 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 #117

Puzzle #117 is one of the hard-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.

Puzzle #117 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 #117 are: slate (3 squares), indigo (4 squares), sea green (5 squares), pale yellow (6 squares), orange (7 squares), amber (11 squares), khaki (12 squares), red (14 squares), mint (17 squares), blue (21 squares), teal (21 squares). An even split would hand every color 11 squares, so the slate region is running 8 squares under that average while the teal region is the roomiest area on the grid with 21 squares.

Every color takes one queen and one only, whatever its size — which is why the 18 square difference between slate and teal 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 1×4 box that encloses it, while red is the most scattered at 44% of its 8×4 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

Below is every color on #117 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.

Slate: 3 squares across rows 5–6, columns 10–11.

Indigo covers 4 squares, spanning 1 row and 4 columns (rows 1–1, columns 3–6). It never leaves row 1, so its queen is locked to that row.

Sea green occupies 5 squares within rows 8–11, columns 1–2.

Pale yellow: 6 squares across rows 5–8, columns 1–2.

Orange covers 7 squares, spanning 3 rows and 3 columns (rows 7–9, columns 4–6).

Amber occupies 11 squares within rows 1–5, columns 1–3.

Khaki: 12 squares across rows 6–11, columns 2–5.

Red covers 14 squares, spanning 8 rows and 4 columns (rows 3–10, columns 8–11).

Mint occupies 17 squares within rows 3–8, columns 3–8.

Blue: 21 squares across rows 1–4, columns 3–11.

Teal covers 21 squares, spanning 5 rows and 8 columns (rows 7–11, columns 4–11).

Where to start on this board

Slate is the natural opening on #117: it is the smallest region on the grid at 3 squares, boxed into rows 5–6 and columns 10–11. Since one of those squares must hold a queen, that block of 2 rows and 2 columns is the most contested part of the board.

Slate does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.

Work outward to the other flat regions next. Indigo is no more than two rows tall, so its queen is 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 #117

A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 2 colors. Columns 1, 7 and 9 are the tightest columns, with 3 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 8 meets 7 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

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.

Keep alternating between two views of the board. In the region view you ask which of the eleven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 11x11 board unwinds.

A placement without a reason is a guess. Because this board has exactly one solution, a guess that happens to be wrong will not announce itself for several moves — and unwinding it is slower than simply finishing the eliminations first.

Why puzzle #117 is rated hard

What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 3-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 18. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

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.

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 #117 is row 2.

As a sanity check rather than a spoiler: the finished solution to #117 puts 4 queens on the outer border and 7 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

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 #117?

Puzzle #117 is rated hard. 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 #117 have only one solution?

Yes. Every puzzle in the 11x11 collection, including #117, 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 #117?

Start with the slate region — the most constrained color on this board, with 3 squares spanning rows 5–6 and columns 10–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.