10x10 Queens Puzzle #89

Difficulty: easy. Place 10 queens on this 10x10 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 21-step reasoning is written out below.

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Step-by-step solution for puzzle #89

What follows is a complete solve of puzzle #89, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 10 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, line confined to one region, pairing argument on columns, region confined to one line.

Show the full walkthrough (15 steps) — contains the solution

  1. Step 1. Only row 3, column 10 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 10, the rest of the blue region and every square adjacent to it — diagonals included.

  2. Step 2. The mint region is down to one open square, at row 4, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 1, the remaining mint squares, and the ring of neighbours around row 4, column 1.

  3. Step 3. Only row 8, column 5 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 5, in khaki and in the ring around that square is now dead.

  4. Step 4. Wherever indigo 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.

  5. Step 5. Pale yellow is down to 3 candidates in rows 6, 7, and 9, columns 7–9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.

    Puzzle #89 after step 5: 3 of 10 queens placed
    The board after step 5 — 3 of 10 queens placed.

  6. Step 6. The orange region has 3 squares left, spread across rows 9 and 10, columns 8 and 9. 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.

  7. Step 7. Wherever pale yellow ends up among its 2 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.

  8. Step 8. Here is the pairing argument. The indigo and amber regions have no open squares outside rows 1 and 2 — 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: 3 squares come off in one stroke.

  9. Step 9. The slate region has 6 squares left, spread across rows 5 and 6, columns 3, 4, 6, 7, 8, and 9. 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.

  10. Step 10. Every open square left in row 5 (4 open squares) belongs to the slate region. Since that row must contain a queen, slate is the colour that supplies it — so slate squares anywhere else on the board are out, 2 squares in all.

    Puzzle #89 after step 10: 3 of 10 queens placed
    The board after step 10 — 3 of 10 queens placed.

  11. Step 11. Orange and sea green are now boxed into rows 9 and 10 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 1 square at once.

  12. Step 12. Here is the pairing argument. The indigo and red regions have no open squares outside columns 2 and 3 — 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.

  13. Step 13. Wherever sea green 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.

  14. Step 14. All of the orange region's remaining squares (2 open squares) lie in column 9. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 3 squares come off.

  15. Step 15. 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: pale yellow has one square left, so its queen takes row 7, column 7; red has one square left, so its queen takes row 6, column 2; indigo has one square left, so its queen takes row 1, column 3; amber has one square left, so its queen takes row 2, column 8; slate has one square left, so its queen takes row 5, column 6; sea green has one square left, so its queen takes row 10, column 4; 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 #89 after step 15: 10 of 10 queens placed
    The board after step 15 — 10 of 10 queens placed.

That completes the board: 10 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 10x10 Queens puzzle #89

Puzzle #89 is one of the easy-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

There is exactly one way to finish #89, 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 ten regions is what sets the depth.

How the colors divide this board

From smallest to largest, the regions on puzzle #89 are: blue (1 square), khaki (1 square), mint (3 squares), indigo (5 squares), orange (5 squares), amber (10 squares), pale yellow (11 squares), sea green (12 squares), red (19 squares), slate (33 squares). An even split would hand every color 10 squares, so the blue region is running 9 squares under that average while the slate region is the roomiest area on the grid with 33 squares.

Every color takes one queen and one only, whatever its size — which is why the 32 square difference between blue and slate is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.

A region's outline can constrain it more than its size does. Here blue is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 55% of its 4×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.

Region by region

Below is every color on #89 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Blue occupies 1 square within rows 3–3, columns 10–10. It never leaves row 3, so its queen is locked to that row.

Khaki: 1 square across rows 8–8, columns 5–5. It never leaves row 8, so its queen is locked to that row.

Mint covers 3 squares, spanning 2 rows and 2 columns (rows 3–4, columns 1–2).

Indigo occupies 5 squares within rows 1–2, columns 1–3.

Orange: 5 squares across rows 9–10, columns 8–10.

Amber covers 10 squares, spanning 2 rows and 7 columns (rows 1–2, columns 4–10).

Pale yellow occupies 11 squares within rows 6–9, columns 6–10.

Sea green: 12 squares across rows 8–10, columns 2–7.

Red covers 19 squares, spanning 8 rows and 4 columns (rows 3–10, columns 1–4).

Slate occupies 33 squares within rows 2–7, columns 3–10.

Where to start on this board

Begin where the choices are fewest. On this board that means blue, which owns just 1 square inside a 1×1 window running from row 3 to row 3 and column 10 to column 10. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.

Because blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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. Khaki, indigo, amber and mint 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 #89

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 5 are the narrowest, touched by only 2 colors. Column 1 is the tightest column, with 3 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 9 meets 4 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

Working through the middle game

The middle of #89 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 ten colors looking for one with a single square left; sweep the ten rows and ten columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 10x10 solve from a stalled one.

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 #89 is rated easy

What earns a board its rating is the shape of its opening. Here that opening offers four single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 32. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

Stuck on #89 despite the easy 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 10x10

At 10x10 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.

There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #89 is row 1.

As a sanity check rather than a spoiler: the finished solution to #89 puts 4 queens on the outer border and 6 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

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 10x10 Queens puzzle #89?

Puzzle #89 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 10x10 puzzle #89 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #89, has been verified to have exactly one valid arrangement of ten queens, and it can always be reached by logic alone — no guessing or backtracking needed.

Where should I place my first queen on puzzle #89?

Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 10–10. Narrowing the smallest region first removes the most squares elsewhere on the grid.