11x11 Queens Puzzle #83

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 24-step reasoning is written out below.

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Step-by-step solution for puzzle #83

What follows is a complete solve of puzzle #83, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 11 queen placements and 13 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, region confined to one line, pairing argument on rows, pairing argument on columns, proof by contradiction.

Show the full walkthrough (18 steps) — contains the solution

  1. Step 1. Only row 1, column 8 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 8, the rest of the indigo region and every square adjacent to it — diagonals included.

  2. Step 2. Blue is down to 2 candidates in rows 3 and 4, columns 7 and 9, and they overlap enough to cover 4 squares elsewhere. Cross those out without deciding anything about blue itself.

  3. Step 3. Pale yellow cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 12 squares in all.

  4. Step 4. Wherever teal 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.

  5. Step 5. Amber, blue, and mint are now boxed into rows 2–4 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 11 squares at once.

    Puzzle #83 after step 5: 1 of 11 queens placed
    The board after step 5 — 1 of 11 queens placed.

  6. Step 6. The slate region is down to one open square, at row 5, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 1, the remaining slate squares, and the ring of neighbours around row 5, column 1.

  7. Step 7. Only row 7, column 2 is still open in pale yellow, and since each colour must take a queen, that settles it. Everything else in row 7, in column 2, in pale yellow and in the ring around that square is now dead.

  8. Step 8. Red is down to 3 candidates in rows 6 and 9, columns 3 and 4, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.

  9. Step 9. Mint and teal are now boxed into columns 10 and 11 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 6 squares at once.

    Puzzle #83 after step 9: 3 of 11 queens placed
    The board after step 9 — 3 of 11 queens placed.

  10. Step 10. Here is the pairing argument. The blue and orange regions have no open squares outside columns 7 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: 3 squares come off in one stroke.

  11. Step 11. Amber and khaki are now boxed into columns 5 and 6 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.

  12. Step 12. Here is the pairing argument. The sea green and teal regions have no open squares outside rows 10 and 11 — 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: 2 squares come off in one stroke.

  13. Step 13. Test row 2, column 6 for amber: 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.

  14. Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 3, column 5. Follow the forced moves that would create and row 8 runs out of open squares entirely — impossible. So that square is ruled out for good.

    Puzzle #83 after step 14: 3 of 11 queens placed
    The board after step 14 — 3 of 11 queens placed.

  15. Step 15. Test row 3, column 6 for amber: 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.

  16. Step 16. The amber region is down to one open square, at row 2, column 5. Every colour holds exactly one queen, so that square is forced. Row 2 and column 5 are now spoken for, along with the squares touching that queen.

  17. Step 17. Test row 4, column 9 for blue: 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.

  18. Step 18. 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: blue has one square left, so its queen takes row 3, column 7; mint has one square left, so its queen takes row 4, column 10; teal has one square left, so its queen takes row 11, column 11; sea green has one square left, so its queen takes row 10, column 3; red has one square left, so its queen takes row 6, column 4; khaki has one square left, so its queen takes row 8, column 6; orange has one square left, so its queen takes row 9, column 9. 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 9, column 9.

    Puzzle #83 after step 18: 11 of 11 queens placed
    The board after step 18 — 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 #83

11x11 puzzle #83 is rated hard. Eleven colored regions cover its 121 squares, and your job is to seat eleven queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Like every board on this site, #83 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #83 are: indigo (1 square), pale yellow (2 squares), teal (3 squares), blue (6 squares), mint (10 squares), amber (12 squares), red (12 squares), slate (14 squares), khaki (16 squares), sea green (21 squares), orange (24 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 24 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 24 to spare, a gap of 23. 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 slate the most diffuse at 50% of its 7×4 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 #83, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Indigo occupies 1 square within rows 1–1, columns 8–8. It never leaves row 1, so its queen is locked to that row.

Pale yellow: 2 squares across rows 6–7, columns 2–2. It sits wholly inside column 2, so its queen must be in that column.

Teal covers 3 squares, spanning 2 rows and 2 columns (rows 10–11, columns 10–11).

Blue occupies 6 squares within rows 2–4, columns 7–9.

Mint: 10 squares across rows 1–4, columns 9–11.

Amber covers 12 squares, spanning 4 rows and 6 columns (rows 1–4, columns 2–7).

Red occupies 12 squares within rows 4–9, columns 3–5.

Slate: 14 squares across rows 1–7, columns 1–4.

Khaki covers 16 squares, spanning 5 rows and 4 columns (rows 4–8, columns 5–8).

Sea green occupies 21 squares within rows 8–11, columns 1–7.

Orange: 24 squares across rows 4–11, columns 7–11.

Where to start on this board

The most constrained color is where a solve should begin, and on #83 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 8 to column 8 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.

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. Pale yellow and teal 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 #83

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 9, 10 and 11 are the narrowest, touched by only 3 colors. Column 1 is the tightest column, with 2 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 4, meeting 7 colors, is the opposite case — the busiest line on #83 and the last one you should expect to crack.

Working through the middle game

Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.

Keep alternating between two views of the board. In the region view you ask which of the eleven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 11x11 board unwinds.

Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.

Why puzzle #83 is rated hard

The rating reflects how much the board concedes up front. Puzzle #83 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 23 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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.

If #83 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

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 column 1 on this board.

Here is a shape check that gives nothing away: when #83 is solved, 3 queens sit on the outer edge of the grid and 8 sit inside it, with 1 queen 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

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. 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.
  5. 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 #83?

Puzzle #83 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 #83 have only one solution?

Yes. Every puzzle in the 11x11 collection, including #83, 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 #83?

Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.