11x11 Queens Puzzle #20
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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #20
What follows is a complete solve of puzzle #20, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 11 queen placements and 9 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 line confined to one region.
Techniques this board needs: last square in a region, region shadow, region confined to one line, line confined to one region, last square in a column.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 2, column 11 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 11, the rest of the amber region and every square adjacent to it — diagonals included.
Step 2. The indigo region is down to one open square, at row 1, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 2, the remaining indigo squares, and the ring of neighbours around row 1, column 2.
Step 3. Only row 11, column 5 is still open in teal, and since each colour must take a queen, that settles it. Everything else in row 11, in column 5, in teal and in the ring around that square is now dead.
Step 4. 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 5. Slate is down to 4 candidates in rows 4 and 5, columns 6–8, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.
The board after step 5 — 3 of 11 queens placed. Step 6. All of the blue region's remaining squares (3 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 3 squares come off.
Step 7. Only row 4, column 1 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 1, in mint and in the ring around that square is now dead.
Step 8. All of the slate region's remaining squares (2 open squares) lie in row 5. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
Step 9. Pale yellow cannot leave column 6 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 4 squares in all.
The board after step 9 — 4 of 11 queens placed. Step 10. Wherever orange ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. Sea green is down to 2 candidates in rows 9 and 10, columns 7 and 8, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about sea green itself.
Step 12. The orange region has 3 squares left, spread across rows 8 and 9, columns 4, 7, and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 13. Column 3 has only red squares left (4 open squares). The column needs a queen and only red can provide one, which pins that colour to this column and kills its squares elsewhere: 2 squares.
Step 14. 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 4 is down to one square, putting a queen on row 9, column 4; sea green has one square left, so its queen takes row 10, column 8; blue has one square left, so its queen takes row 3, column 9; slate has one square left, so its queen takes row 5, column 7; pale yellow has one square left, so its queen takes row 7, column 6; red has one square left, so its queen takes row 6, column 3; khaki has one square left, so its queen takes row 8, column 10. 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 8, column 10.
The board after step 14 — 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 #20
This is puzzle number 20 in the 11x11 collection, rated medium. The grid holds 121 squares split between eleven colored regions, and a finished board carries eleven queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #20, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these eleven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #20 are: amber (1 square), teal (1 square), indigo (2 squares), pale yellow (5 squares), slate (6 squares), sea green (8 squares), mint (11 squares), red (20 squares), khaki (20 squares), orange (23 squares), blue (24 squares). An even split would hand every color 11 squares, so the amber region is running 10 squares under that average while the blue region is the roomiest area on the grid with 24 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 1 square while blue has 24 to spare, a gap of 23. Attack the cramped end and the roomy regions tend to resolve themselves.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 2×1 box, and red the most diffuse at 48% of its 7×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Amber occupies 1 square within rows 2–2, columns 11–11. It never leaves row 2, so its queen is locked to that row.
Teal: 1 square across rows 11–11, columns 5–5. It never leaves row 11, so its queen is locked to that row.
Indigo covers 2 squares, spanning 2 rows and 1 column (rows 1–2, columns 2–2). It sits wholly inside column 2, so its queen must be in that column.
Pale yellow occupies 5 squares within rows 6–7, columns 5–8.
Slate: 6 squares across rows 3–5, columns 5–8.
Sea green covers 8 squares, spanning 3 rows and 5 columns (rows 9–11, columns 6–10).
Mint occupies 11 squares within rows 1–5, columns 1–4.
Red: 20 squares across rows 4–10, columns 1–6.
Khaki covers 20 squares, spanning 8 rows and 4 columns (rows 4–11, columns 8–11).
Orange occupies 23 squares within rows 7–11, columns 1–9.
Blue: 24 squares across rows 1–5, columns 4–11.
Where to start on this board
Open on the tightest color. Here that is amber, confined to 1 square between rows 2 and 2 and columns 11 and 11. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #20.
Because amber never leaves row 2, its queen must be in that row — and that instantly settles row 2 for everyone else. Sweep along it and X out every square that is not amber. 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, indigo and pale yellow 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 #20
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 6 and 8 are the narrowest, touched by only 3 colors. Columns 1, 3, 10 and 11 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 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 10 rows. By contrast row 5 meets 5 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.
Two checks, run alternately, will carry you through the 121 squares of this grid. First: has any of the eleven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 11x11 boards almost always run one of those checks and forget the other.
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 #20 is rated medium
The medium label comes out of four measurable things about #20: five 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 23. 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.
Stuck on #20 despite the medium rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 row 1 on this board.
As a sanity check rather than a spoiler: the finished solution to #20 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.
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Questions about this puzzle
How hard is 11x11 Queens puzzle #20?
Puzzle #20 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and five of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #20 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #20, 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 #20?
Start with the amber region — the most constrained color on this board, with 1 square spanning rows 2–2 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.