11x11 Queens Puzzle #172
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 #172
What follows is a complete solve of puzzle #172, 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: last square in a region, region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, last square in a row, line confined to one region.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 6, column 5 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 5, the rest of the red region and every square adjacent to it — diagonals included.
Step 2. All of the amber 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: 10 squares come off.
Step 3. The mint region has 4 squares left, spread across rows 2–4, columns 9 and 10. 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 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.
The board after step 4 — 1 of 11 queens placed. Step 5. Indigo and amber are now boxed into rows 1 and 2 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 12 squares at once.
Step 6. Here is the pairing argument. The blue, mint, and slate regions have no open squares outside rows 3–5 — 3 colours confined to 3 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 7. Wherever pale yellow 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 8. Here is the pairing argument. The mint and teal regions have no open squares outside columns 9 and 10 — 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 9. Indigo and khaki are now boxed into columns 7 and 8 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 10 squares at once.
The board after step 9 — 1 of 11 queens placed. Step 10. Row 5 has a single open square left, at column 1 in the slate region. Every row needs a queen, so this one is forced. Row 5 and column 1 are now spoken for, along with the squares touching that queen.
Step 11. Look along row 4: one square survives, row 4, column 10, which belongs to mint. A row cannot be left empty, so the queen goes there. That clears row 4, column 10, the rest of the mint region and every square adjacent to it — diagonals included.
Step 12. The teal region is down to one open square, at row 11, column 9. Every colour holds exactly one queen, so that square is forced. Row 11 and column 9 are now spoken for, along with the squares touching that queen.
Step 13. Wherever orange 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.
The board after step 13 — 4 of 11 queens placed. Step 14. Every open square left in column 2 (2 open squares) belongs to the orange region. Since that column must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 1 square in all.
Step 15. Orange cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 1 square in all.
Step 16. Every open square left in row 10 (2 open squares) belongs to the sea green region. Since that row must contain a queen, sea green is the colour that supplies it — so sea green squares anywhere else on the board are out, 3 squares in all.
Step 17. 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: row 9 is down to one square, putting a queen on row 9, column 2; sea green has one square left, so its queen takes row 10, column 6; blue has one square left, so its queen takes row 3, column 4; pale yellow has one square left, so its queen takes row 7, column 3; khaki has one square left, so its queen takes row 8, column 8; indigo has one square left, so its queen takes row 1, column 7; amber has one square left, so its queen takes row 2, column 11. 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 2, column 11.
The board after step 17 — 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 #172
This is puzzle number 172 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 #172, 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 #172 are: red (1 square), amber (2 squares), indigo (5 squares), teal (7 squares), pale yellow (9 squares), mint (10 squares), orange (12 squares), blue (17 squares), slate (18 squares), khaki (19 squares), sea green (21 squares). An even split would hand every color 11 squares, so the red region is running 10 squares under that average while the sea green region is the roomiest area on the grid with 21 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Red is working with 1 square while sea green has 21 to spare, a gap of 20. 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 amber is the most solid, filling 100% of its 2×1 box, and blue the most diffuse at 47% of its 6×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #172, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Red covers 1 square, spanning 1 row and 1 column (rows 6–6, columns 5–5). It never leaves row 6, so its queen is locked to that row.
Amber occupies 2 squares within rows 1–2, columns 11–11. It sits wholly inside column 11, so its queen must be in that column.
Indigo: 5 squares across rows 1–2, columns 7–10.
Teal covers 7 squares, spanning 3 rows and 3 columns (rows 9–11, columns 9–11).
Pale yellow occupies 9 squares within rows 5–8, columns 1–4.
Mint: 10 squares across rows 2–6, columns 9–11.
Orange covers 12 squares, spanning 5 rows and 4 columns (rows 7–11, columns 1–4).
Blue occupies 17 squares within rows 1–6, columns 3–8.
Slate: 18 squares across rows 1–6, columns 1–6.
Khaki covers 19 squares, spanning 7 rows and 5 columns (rows 4–10, columns 7–11).
Sea green occupies 21 squares within rows 6–11, columns 2–8.
Where to start on this board
Open on the tightest color. Here that is red, confined to 1 square between rows 6 and 6 and columns 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #172.
Because red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Indigo and amber 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 #172
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 11 are the narrowest, touched by only 3 colors. Columns 1 and 6 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 10 rows. Row 6, meeting 7 colors, is the opposite case — the busiest line on #172 and the last one you should expect to crack.
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 #172 is rated hard
The rating reflects how much the board concedes up front. Puzzle #172 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 20 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
Stuck on #172 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.
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 3 on this board.
Here is a shape check that gives nothing away: when #172 is solved, 4 queens sit on the outer edge of the grid and 7 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #172?
Puzzle #172 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #172 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #172, 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 #172?
Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.