11x11 Queens Puzzle #176
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 #176
What follows is a complete solve of puzzle #176, 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 proof by contradiction.
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, proof by contradiction.
Show the full walkthrough (22 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 sea green region is down to one open square, at row 10, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 6, the remaining sea green squares, and the ring of neighbours around row 10, column 6.
Step 3. The khaki region has 4 squares left, spread across rows 8 and 9, columns 1–3. 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. Here is the pairing argument. The pale yellow, khaki, orange, and teal regions have no open squares outside rows 7, 8, 9, and 11 — 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: 8 squares come off in one stroke.
Step 5. Red is down to 3 candidates in rows 4–6, columns 5 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 6. Here is the pairing argument. The amber, pale yellow, and khaki regions have no open squares outside columns 1–3 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 10 squares come off in one stroke.
The board after step 6 — 2 of 11 queens placed. Step 7. Wherever blue 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. All of the teal region's remaining squares (2 open squares) lie in column 4. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 9. Only row 3, column 5 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 5, the rest of the blue region and every square adjacent to it — diagonals included.
Step 10. The red region is down to one open square, at row 6, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 7, the remaining red squares, and the ring of neighbours around row 6, column 7.
Step 11. Only row 4, column 9 is still open in mint, and since each colour must take a queen, that settles it. Everything else in row 4, in column 9, in mint and in the ring around that square is now dead.
The board after step 11 — 5 of 11 queens placed. Step 12. Every open square left in row 7 (3 open squares) belongs to the pale yellow region. Since that row must contain a queen, pale yellow is the colour that supplies it — so pale yellow squares anywhere else on the board are out, 1 square in all.
Step 13. Test row 2, column 1 for amber: place a queen there, run the forced consequences through, and column 3 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 5, column 1. Follow the forced moves that would create and column 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Only row 2, column 2 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 2, in amber and in the ring around that square is now dead.
Step 16. The slate region is down to one open square, at row 5, column 11. Every colour holds exactly one queen, so that square is forced. Row 5 and column 11 are now spoken for, along with the squares touching that queen.
Step 17. Test row 7, column 3 for pale yellow: place a queen there, run the forced consequences through, and row 8 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 17 — 7 of 11 queens placed. Step 18. The pale yellow region is down to one open square, at row 7, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 1, the remaining pale yellow squares, and the ring of neighbours around row 7, column 1.
Step 19. Khaki cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 1 square in all.
Step 20. The teal region is down to one open square, at row 11, column 4. Every colour holds exactly one queen, so that square is forced. Row 11 and column 4 are now spoken for, along with the squares touching that queen.
Step 21. Only row 9, column 8 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 8, the rest of the orange region and every square adjacent to it — diagonals included.
Step 22. 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.
The board after step 22 — 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 #176
This is puzzle number 176 in the 11x11 collection, rated easy. 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 #176 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 #176 are: indigo (1 square), sea green (1 square), khaki (4 squares), pale yellow (5 squares), red (9 squares), teal (12 squares), amber (14 squares), slate (15 squares), blue (17 squares), mint (20 squares), orange (23 squares). An even split would hand every color 11 squares, so the indigo region is running 10 squares under that average while the orange region is the roomiest area on the grid with 23 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while orange has 23 to spare, a gap of 22. 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 1×1 box, and mint the most diffuse at 44% of its 5×9 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 easy board.
Indigo occupies 1 square within rows 1–1, columns 10–10. It never leaves row 1, so its queen is locked to that row.
Sea green: 1 square across rows 10–10, columns 6–6. It never leaves row 10, so its queen is locked to that row.
Khaki covers 4 squares, spanning 2 rows and 3 columns (rows 8–9, columns 1–3).
Pale yellow occupies 5 squares within rows 7–8, columns 1–3.
Red: 9 squares across rows 4–8, columns 5–8.
Teal covers 12 squares, spanning 4 rows and 5 columns (rows 8–11, columns 1–5).
Amber occupies 14 squares within rows 1–5, columns 1–6.
Slate: 15 squares across rows 2–8, columns 8–11.
Blue covers 17 squares, spanning 6 rows and 6 columns (rows 3–8, columns 1–6).
Mint occupies 20 squares within rows 1–5, columns 3–11.
Orange: 23 squares across rows 7–11, columns 5–11.
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 #176.
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. Sea green, pale yellow 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 #176
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 7, 9 and 11 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 10 rows. Row 8 is the crowded extreme at 7 colors — worth revisiting later, not now.
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 #176 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 #176 is rated easy
The easy label comes out of four measurable things about #176: four of its regions are locked to one row or column, its smallest color spans 1 square, 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 easy board from the rest.
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.
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.
A non-spoiling distribution check: solved, #176 carries 4 queens on the perimeter against 7 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
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 #176?
Puzzle #176 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 11x11 puzzle #176 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #176, 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 #176?
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.