11x11 Queens Puzzle #28
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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #28
What follows is a complete solve of puzzle #28, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 11 queen placements and 5 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 (9 steps) — contains the solution
Step 1. Only row 1, column 10 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 10, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The khaki region is down to one open square, at row 8, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 3, the remaining khaki squares, and the ring of neighbours around row 8, column 3.
Step 3. Amber cannot leave column 1 — every one of its open squares (3 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 7 squares in all.
The board after step 3 — 2 of 11 queens placed. Step 4. All of the mint region's remaining squares (2 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 6 squares come off.
Step 5. Only row 10, column 9 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 9, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 6. The slate region has 4 squares left, spread across rows 2, 3, and 5, columns 7 and 8. 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.
The board after step 6 — 3 of 11 queens placed. Step 7. Wherever teal 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 8. Here is the pairing argument. The amber, blue, mint, and slate regions have no open squares outside rows 2–5 — 4 colours confined to 4 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 7 squares come off in one stroke.
Step 9. 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: red has one square left, so its queen takes row 6, column 2; pale yellow has one square left, so its queen takes row 7, column 6; column 4 is down to one square, putting a queen on row 11, column 4; orange has one square left, so its queen takes row 9, column 7; blue has one square left, so its queen takes row 3, column 5; mint has one square left, so its queen takes row 4, column 11; amber has one square left, so its queen takes row 2, column 1; slate has one square left, so its queen takes row 5, 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 5, column 8.
The board after step 9 — 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 #28
This is puzzle number 28 in the 11x11 collection, rated hard. 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.
Puzzle #28 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 #28 are: indigo (1 square), khaki (1 square), amber (5 squares), pale yellow (7 squares), mint (8 squares), sea green (9 squares), teal (11 squares), orange (17 squares), blue (20 squares), slate (20 squares), red (22 squares). An even split would hand every color 11 squares, so the indigo 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 indigo 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.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and red the most diffuse at 44% of its 10×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 10–10). It never leaves row 1, so its queen is locked to that row.
Khaki occupies 1 square within rows 8–8, columns 3–3. It never leaves row 8, so its queen is locked to that row.
Amber: 5 squares across rows 1–4, columns 1–2.
Pale yellow covers 7 squares, spanning 3 rows and 4 columns (rows 5–7, columns 4–7).
Mint occupies 8 squares within rows 1–5, columns 10–11.
Sea green: 9 squares across rows 7–11, columns 9–11.
Teal covers 11 squares, spanning 5 rows and 4 columns (rows 7–11, columns 2–5).
Orange occupies 17 squares within rows 6–11, columns 5–9.
Blue: 20 squares across rows 1–5, columns 2–8.
Slate covers 20 squares, spanning 9 rows and 5 columns (rows 1–9, columns 7–11).
Red occupies 22 squares within rows 2–11, columns 1–5.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 10 and 10. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #28.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Khaki 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 #28
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6, 10 and 11 are the narrowest, touched by only 4 colors. Column 1 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 4 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 8 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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.
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 #28 is rated hard
Ratings track how generous the opening is. On #28 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 4 colors, and a 21-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
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 #28 is column 1.
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 #28?
Puzzle #28 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 4 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #28 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #28, 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 #28?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.