11x11 Queens Puzzle #36
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 23-step reasoning is written out below.
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Step-by-step solution for puzzle #36
What follows is a complete solve of puzzle #36, worked the way a person works it rather than the way a computer does. It runs to 23 deductions — 11 queen placements and 12 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 columns.
Techniques this board needs: region confined to one line, region shadow, pairing argument on columns, line confined to one region, last square in a region, pairing argument on rows.
Show the full walkthrough (16 steps) — contains the solution
Step 1. 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: 9 squares in all.
Step 2. All of the amber region's remaining squares (3 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
Step 3. The blue region has 3 squares left, spread across rows 3 and 4, columns 6 and 7. 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 4. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 5. Khaki is down to 3 candidates in rows 7 and 8, columns 6 and 7, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about khaki itself.
The board after step 5 — 0 of 11 queens placed. Step 6. Here is the pairing argument. The blue and khaki regions have no open squares outside columns 6 and 7 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 7. Wherever slate 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 8. Here is the pairing argument. The indigo and red regions have no open squares outside columns 10 and 11 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 11 squares come off in one stroke.
Step 9. Row 11 has only teal squares left (7 open squares). The row needs a queen and only teal can provide one, which pins that colour to this row and kills its squares elsewhere: 9 squares.
Step 10. Here is the pairing argument. The amber and orange regions have no open squares outside columns 8 and 9 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 6 squares come off in one stroke.
Step 11. Only row 5, column 5 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 5, in slate and in the ring around that square is now dead.
The board after step 11 — 1 of 11 queens placed. Step 12. Here is the pairing argument. The blue, mint, red, and pale yellow regions have no open squares outside rows 3, 4, 6, and 7 — 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: 5 squares come off in one stroke.
Step 13. Only row 8, column 7 is still open in khaki, and since each colour must take a queen, that settles it. That clears row 8, column 7, the rest of the khaki region and every square adjacent to it — diagonals included.
Step 14. The blue region is down to one open square, at row 3, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 6, the remaining blue squares, and the ring of neighbours around row 3, column 6.
Step 15. Sea green cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 4 squares in all.
Step 16. 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: mint has one square left, so its queen takes row 4, column 2; red has one square left, so its queen takes row 6, column 11; indigo has one square left, so its queen takes row 1, column 10; amber has one square left, so its queen takes row 2, column 8; pale yellow has one square left, so its queen takes row 7, column 4; orange has one square left, so its queen takes row 9, column 9; sea green has one square left, so its queen takes row 10, column 1; teal has one square left, so its queen takes row 11, 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 11, column 3.
The board after step 16 — 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 #36
This is puzzle number 36 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.
As with every puzzle in this collection, #36 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #36 are: indigo (3 squares), khaki (3 squares), amber (5 squares), sea green (6 squares), blue (8 squares), red (9 squares), slate (11 squares), mint (13 squares), orange (20 squares), teal (21 squares), pale yellow (22 squares). An even split would hand every color 11 squares, so the indigo region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 22 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 3 squares while pale yellow has 22 to spare, a gap of 19. 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 1×3 box, and sea green the most diffuse at 50% of its 4×3 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: 3 squares across rows 1–1, columns 9–11. It never leaves row 1, so its queen is locked to that row.
Khaki covers 3 squares, spanning 2 rows and 2 columns (rows 7–8, columns 6–7).
Amber occupies 5 squares within rows 1–2, columns 7–10.
Sea green: 6 squares across rows 7–10, columns 1–3.
Blue covers 8 squares, spanning 4 rows and 4 columns (rows 1–4, columns 4–7).
Red occupies 9 squares within rows 2–6, columns 9–11.
Slate: 11 squares across rows 3–6, columns 5–9.
Mint covers 13 squares, spanning 7 rows and 3 columns (rows 1–7, columns 1–3).
Orange occupies 20 squares within rows 6–11, columns 7–11.
Teal: 21 squares across rows 8–11, columns 1–9.
Pale yellow covers 22 squares, spanning 8 rows and 5 columns (rows 2–9, columns 2–6).
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 3 squares between rows 1 and 1 and columns 9 and 11. Its queen has nowhere else to go, which means those 1 row and 3 columns carry more constraint than anywhere else on #36.
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. Amber and khaki 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 #36
A quick way to find the next move is to count how many different colors touch each line. On this board, row 11 is the narrowest, touched by only 2 colors. Columns 1, 4 and 11 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 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 5 colors crossing it; it will almost certainly be among the last to settle.
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.
Resist the plausible-looking queen. On #36 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #36 is rated hard
Ratings track how generous the opening is. On #36 you get one region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 2 colors, and a 19-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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 11 on this board.
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 #36?
Puzzle #36 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 #36 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #36, 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 #36?
Start with the indigo region — the most constrained color on this board, with 3 squares spanning rows 1–1 and columns 9–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.