11x11 Queens Puzzle #52
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 #52
What follows is a complete solve of puzzle #52, 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, last square in a region, pairing argument on rows, pairing argument on columns.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Amber cannot leave column 11 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 11 squares in all.
Step 2. Slate is down to 6 candidates in rows 3–5, columns 8–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
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. Wherever sea green ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 5. Teal cannot leave row 11 — every one of its open squares (7 open squares) sits there. That row belongs to teal, so every square on it owned by another colour is dead: 2 squares in all.
The board after step 5 — 0 of 11 queens placed. Step 6. 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: sea green has one square left, so its queen takes row 10, column 9; indigo has one square left, so its queen takes row 1, column 8; amber has one square left, so its queen takes row 2, column 11; blue has one square left, so its queen takes row 3, column 1. 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 3, column 1.
Step 7. Wherever mint 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. Slate cannot leave column 10 — 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.
Step 9. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
The board after step 9 — 4 of 11 queens placed. Step 10. Orange cannot leave row 9 — every one of its open squares (4 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 2 squares in all.
Step 11. Here is the pairing argument. The mint and slate regions have no open squares outside rows 4 and 5 — 2 colours confined to 2 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 12. The red region has 3 squares left, spread across rows 6 and 8, columns 4 and 5. 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 13. Here is the pairing argument. The mint and pale yellow 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: 4 squares come off in one stroke.
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: orange has one square left, so its queen takes row 9, column 4; red has one square left, so its queen takes row 6, column 5; pale yellow has one square left, so its queen takes row 7, column 7; mint has one square left, so its queen takes row 4, column 6; slate has one square left, so its queen takes row 5, column 10; khaki has one square left, so its queen takes row 8, column 2; 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 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 #52
This is puzzle number 52 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.
There is exactly one way to finish #52, 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 #52 are: amber (2 squares), sea green (6 squares), indigo (7 squares), teal (8 squares), orange (9 squares), slate (10 squares), blue (11 squares), mint (12 squares), khaki (12 squares), pale yellow (20 squares), red (24 squares). An even split would hand every color 11 squares, so the amber region is running 9 squares under that average while the red region is the roomiest area on the grid with 24 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Amber has only 2 places to put its queen; red has 24. The spread between them on this board is 22 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
A region's outline can constrain it more than its size does. Here amber is the most solid, filling 100% of its 2×1 box, and red the most diffuse at 49% of its 7×7 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 hard board.
Amber covers 2 squares, spanning 2 rows and 1 column (rows 1–2, columns 11–11). It sits wholly inside column 11, so its queen must be in that column.
Sea green occupies 6 squares within rows 9–11, columns 9–11.
Indigo: 7 squares across rows 1–4, columns 8–10.
Teal covers 8 squares, spanning 1 row and 8 columns (rows 11–11, columns 1–8). It never leaves row 11, so its queen is locked to that row.
Orange occupies 9 squares within rows 9–10, columns 1–7.
Slate: 10 squares across rows 3–6, columns 8–11.
Blue covers 11 squares, spanning 3 rows and 7 columns (rows 1–3, columns 1–7).
Mint occupies 12 squares within rows 2–6, columns 5–8.
Khaki: 12 squares across rows 6–9, columns 1–4.
Pale yellow covers 20 squares, spanning 5 rows and 6 columns (rows 6–10, columns 6–11).
Red occupies 24 squares within rows 2–8, columns 1–7.
Where to start on this board
Open on the tightest color. Here that is amber, confined to 2 squares between rows 1 and 2 and columns 11 and 11. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #52.
Because amber stays inside column 11, its queen belongs to that column and no other region may use it. Cross out the rest of column 11 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.
Work outward to the other flat regions next. Teal and orange 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 #52
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 9, 10 and 11 are the tightest columns, with 4 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 6 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
The middle of #52 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
The whole middle game reduces to two sweeps. Sweep the eleven colors looking for one with a single square left; sweep the eleven rows and eleven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 11x11 solve from a stalled one.
Resist the plausible-looking queen. On #52 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 #52 is rated hard
The hard label comes out of four measurable things about #52: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 22. Boards that score tight on those measures give up their first queen fast.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a hard board from the rest.
Stuck on #52 despite the hard 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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #52 the place to look for one is around row 11.
As a sanity check rather than a spoiler: the finished solution to #52 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 #52?
Puzzle #52 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #52 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #52, 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 #52?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.