11x11 Queens Puzzle #146
Difficulty: easy. 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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #146
What follows is a complete solve of puzzle #146, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 11 queen placements and 11 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: last square in a region, region shadow, pairing argument on rows, pairing argument on columns, region confined to one line, line confined to one region, last square in a column.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 1, column 11 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 11, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The orange region is down to one open square, at row 9, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 1, the remaining orange squares, and the ring of neighbours around row 9, column 1.
Step 3. The khaki region has 3 squares left, spread across rows 7 and 8, columns 9 and 10. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 4. Wherever teal ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Sea green and teal are now boxed into rows 10 and 11 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 8 squares at once.
The board after step 5 — 2 of 11 queens placed. Step 6. The pale yellow region has 4 squares left, spread across rows 7 and 8, columns 3–5. 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 7. Wherever slate 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 pale yellow and khaki regions have no open squares outside rows 7 and 8 — 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: 6 squares come off in one stroke.
Step 9. The slate region has 3 squares left, spread across rows 5 and 6, columns 2 and 3. 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 10. Here is the pairing argument. The slate and red regions have no open squares outside rows 5 and 6 — 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 11. Slate and teal are now boxed into columns 2 and 3 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 7 squares at once.
The board after step 11 — 2 of 11 queens placed. Step 12. All of the amber region's remaining squares (3 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.
Step 13. Only row 7, column 5 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 5, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 14. The khaki region is down to one open square, at row 8, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 9, the remaining khaki squares, and the ring of neighbours around row 8, column 9.
Step 15. Column 6 has only mint squares left (3 open squares). The column needs a queen and only mint can provide one, which pins that colour to this column and kills its squares elsewhere: 5 squares.
Step 16. 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 7 is down to one square, putting a queen on row 6, column 7; slate has one square left, so its queen takes row 5, column 3; teal has one square left, so its queen takes row 11, column 2; sea green has one square left, so its queen takes row 10, column 10; blue has one square left, so its queen takes row 3, column 8; amber has one square left, so its queen takes row 2, column 4; mint has one square left, so its queen takes row 4, column 6. 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 4, column 6.
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 #146
Board #146 of the 11x11 bank carries a easy rating. Its 121 squares are shared between eleven colored regions, and the solution places eleven queens: one per row, one per column, one per region, none of them side by side or corner to corner.
As with every puzzle in this collection, #146 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 #146 are: indigo (1 square), orange (1 square), khaki (6 squares), teal (7 squares), slate (9 squares), sea green (12 squares), pale yellow (14 squares), mint (15 squares), blue (18 squares), red (18 squares), amber (20 squares). An even split would hand every color 11 squares, so the indigo region is running 10 squares under that average while the amber region is the roomiest area on the grid with 20 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So indigo, with 1 square, is under far more pressure than amber with 20. A 19-square gap like this one rewards players who start where the choices are fewest.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; red is the loosest, occupying only 43% of its 7×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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 easy board.
Indigo occupies 1 square within rows 1–1, columns 11–11. It never leaves row 1, so its queen is locked to that row.
Orange: 1 square across rows 9–9, columns 1–1. It never leaves row 9, so its queen is locked to that row.
Khaki covers 6 squares, spanning 3 rows and 3 columns (rows 6–8, columns 9–11).
Teal occupies 7 squares within rows 9–11, columns 1–3.
Slate: 9 squares across rows 5–8, columns 1–3.
Sea green covers 12 squares, spanning 4 rows and 4 columns (rows 8–11, columns 8–11).
Pale yellow occupies 14 squares within rows 7–11, columns 3–6.
Mint: 15 squares across rows 2–5, columns 4–8.
Blue covers 18 squares, spanning 6 rows and 5 columns (rows 1–6, columns 7–11).
Red occupies 18 squares within rows 5–11, columns 4–9.
Amber: 20 squares across rows 1–5, columns 1–6.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 11 to column 11. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
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. Orange 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 #146
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3 and 4 are the narrowest, touched by only 3 colors. Columns 2, 7 and 10 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 3 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 9 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
The middle of #146 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.
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 #146 is rated easy
The easy label comes out of four measurable things about #146: four 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 19. Boards that score tight on those measures give up their first queen fast.
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.
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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #146, start hunting for it near row 1.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 8 queens in the middle, including 1 in a corner. 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 #146?
Puzzle #146 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #146 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #146, 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 #146?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 11–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.