10x10 Queens Puzzle #55

Difficulty: hard. 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 17-step reasoning is written out below.

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Step-by-step solution for puzzle #55

What follows is a complete solve of puzzle #55, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 10 queen placements and 7 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, region confined to one line, pairing argument on rows, pairing argument on columns.

Show the full walkthrough (10 steps) — contains the solution

  1. 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.

  2. Step 2. 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.

  3. Step 3. The amber region has 3 squares left, spread across rows 2 and 3, columns 2 and 4. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.

    Puzzle #55 after step 3: 2 of 10 queens placed
    The board after step 3 — 2 of 10 queens placed.

  4. Step 4. Wherever blue 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.

  5. Step 5. Slate cannot leave column 2 — 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: 6 squares in all.

  6. Step 6. All of the amber 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: 7 squares come off.

  7. Step 7. Amber and blue are now boxed into rows 2 and 3 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 3 squares at once.

    Puzzle #55 after step 7: 2 of 10 queens placed
    The board after step 7 — 2 of 10 queens placed.

  8. Step 8. Mint is down to 2 candidates in rows 4 and 6, columns 3 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.

  9. Step 9. Mint and orange are now boxed into columns 3 and 5 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 2 squares at once.

  10. Step 10. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 6; mint has one square left, so its queen takes row 4, column 5; amber has one square left, so its queen takes row 2, column 4; blue has one square left, so its queen takes row 3, column 9; slate has one square left, so its queen takes row 5, column 2; khaki has one square left, so its queen takes row 8, column 7; sea green has one square left, so its queen takes row 10, column 8; orange has one square left, so its queen takes row 9, column 3. 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 3.

    Puzzle #55 after step 10: 10 of 10 queens placed
    The board after step 10 — 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 #55

10x10 puzzle #55 is rated hard. Ten colored regions cover its 100 squares, and your job is to seat ten queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.

Like every board on this site, #55 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 #55 are: indigo (1 square), pale yellow (1 square), red (8 squares), amber (9 squares), blue (9 squares), slate (10 squares), sea green (11 squares), orange (14 squares), mint (15 squares), khaki (22 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 22 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; khaki has 22. The spread between them on this board is 21 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; mint is the loosest, occupying only 42% of its 6×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

Region by region

Here is the full footprint of every color on puzzle #55, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 10–10). It never leaves row 1, so its queen is locked to that row.

Pale yellow occupies 1 square within rows 7–7, columns 1–1. It never leaves row 7, so its queen is locked to that row.

Red: 8 squares across rows 5–8, columns 4–6.

Amber covers 9 squares, spanning 3 rows and 5 columns (rows 1–3, columns 1–5).

Blue occupies 9 squares within rows 1–4, columns 7–10.

Slate: 10 squares across rows 3–7, columns 1–4.

Sea green covers 11 squares, spanning 3 rows and 4 columns (rows 8–10, columns 7–10).

Orange occupies 14 squares within rows 8–10, columns 1–5.

Mint: 15 squares across rows 1–6, columns 2–7.

Khaki covers 22 squares, spanning 8 rows and 5 columns (rows 3–10, columns 6–10).

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 #55.

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 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 #55

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 9 and 10 are the narrowest, touched by only 3 colors. Columns 6, 8 and 9 are the tightest columns, with 3 colors.

A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 9 rows. Row 3 is the crowded extreme at 5 colors — worth revisiting later, not now.

Working through the middle game

The middle of #55 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.

Keep alternating between two views of the board. In the region view you ask which of the ten 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 10x10 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 #55 is rated hard

The rating reflects how much the board concedes up front. Puzzle #55 has four 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 21 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.

If #55 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 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.

One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #55 the place to look for one is around row 2.

A non-spoiling distribution check: solved, #55 carries 3 queens on the perimeter against 7 queens inside, 1 of them cornered. 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

  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 #55?

Puzzle #55 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #55 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #55, 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 #55?

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.