11x11 Queens Puzzle #216

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 19-step reasoning is written out below.

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Step-by-step solution for puzzle #216

What follows is a complete solve of puzzle #216, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 11 queen placements and 8 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 rows.

Techniques this board needs: region confined to one line, region shadow, pairing argument on rows, last square in a region.

Show the full walkthrough (14 steps) — contains the solution

  1. Step 1. Indigo cannot leave column 11 — every one of its open squares (2 open squares) sits there. That column belongs to indigo, so every square in it owned by another colour is dead: 11 squares in all.

  2. Step 2. Slate is down to 6 candidates in rows 3–6, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.

  3. Step 3. Pale yellow cannot leave column 7 — every one of its open squares (3 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 10 squares in all.

  4. Step 4. Here is the pairing argument. The indigo, amber, blue, and mint regions have no open squares outside rows 1–4 — 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: 9 squares come off in one stroke.

  5. Step 5. Slate cannot leave column 1 — 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: 9 squares in all.

    Puzzle #216 after step 5: 0 of 11 queens placed
    The board after step 5 — 0 of 11 queens placed.

  6. Step 6. The amber region is down to one open square, at row 2, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 2, the remaining amber squares, and the ring of neighbours around row 2, column 2.

  7. Step 7. Only row 1, column 11 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 11, in indigo and in the ring around that square is now dead.

  8. Step 8. Mint is down to 3 candidates in rows 3 and 4, columns 4 and 5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about mint itself.

  9. Step 9. Slate, red, and pale yellow are now boxed into rows 5–7 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 #216 after step 9: 2 of 11 queens placed
    The board after step 9 — 2 of 11 queens placed.

  10. Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: khaki has one square left, so its queen takes row 8, column 10; blue has one square left, so its queen takes row 3, column 6; mint has one square left, so its queen takes row 4, column 4; red has one square left, so its queen takes row 6, column 5; slate has one square left, so its queen takes row 5, column 1; pale yellow has one square left, so its queen takes row 7, column 7. 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 7, column 7.

  11. Step 11. All of the orange region's remaining squares (2 open squares) lie in column 8. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.

  12. Step 12. Only row 11, column 9 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 9, the rest of the teal region and every square adjacent to it — diagonals included.

  13. Step 13. The orange region is down to one open square, at row 9, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 8, the remaining orange squares, and the ring of neighbours around row 9, column 8.

  14. Step 14. Only row 10, column 3 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 3, in sea green and in the ring around that square is now dead.

    Puzzle #216 after step 14: 11 of 11 queens placed
    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 #216

This is puzzle number 216 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 #216, 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 #216 are: indigo (2 squares), pale yellow (3 squares), amber (4 squares), slate (6 squares), teal (9 squares), mint (11 squares), red (12 squares), orange (12 squares), blue (18 squares), khaki (19 squares), sea green (25 squares). An even split would hand every color 11 squares, so the indigo region is running 9 squares under that average while the sea green region is the roomiest area on the grid with 25 squares.

One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 2 squares while sea green has 25 to spare, a gap of 23. Attack the cramped end and the roomy regions tend to resolve themselves.

Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 2×1 box that encloses it, while orange is the most scattered at 50% of its 4×6 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.

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.

Indigo: 2 squares across rows 1–2, columns 11–11. It sits wholly inside column 11, so its queen must be in that column.

Pale yellow covers 3 squares, spanning 3 rows and 1 column (rows 5–7, columns 7–7). It sits wholly inside column 7, so its queen must be in that column.

Amber occupies 4 squares within rows 1–3, columns 1–2.

Slate: 6 squares across rows 3–6, columns 1–3.

Teal covers 9 squares, spanning 2 rows and 5 columns (rows 10–11, columns 7–11).

Mint occupies 11 squares within rows 1–4, columns 2–5.

Red: 12 squares across rows 4–7, columns 2–6.

Orange covers 12 squares, spanning 4 rows and 6 columns (rows 8–11, columns 5–10).

Blue occupies 18 squares within rows 1–4, columns 5–11.

Khaki: 19 squares across rows 3–9, columns 8–11.

Sea green covers 25 squares, spanning 6 rows and 6 columns (rows 6–11, columns 1–6).

Where to start on this board

Open on the tightest color. Here that is indigo, 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 #216.

Because indigo 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 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 #216

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 8, 9, 10 and 11 are the narrowest, touched by only 3 colors. Columns 1 and 4 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 5 colors, is the opposite case — the busiest line on #216 and the last one you should expect to crack.

Working through the middle game

The middle of #216 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.

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 #216 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 #216 is rated hard

The hard label comes out of four measurable things about #216: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 23. Boards that score tight on those measures give up their first queen fast.

Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.

Stuck on #216 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 8 on this board.

Here is a shape check that gives nothing away: when #216 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 #216?

Puzzle #216 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 11x11 puzzle #216 have only one solution?

Yes. Every puzzle in the 11x11 collection, including #216, 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 #216?

Start with the indigo 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.