10x10 Queens Puzzle #22

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 16-step reasoning is written out below.

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Step-by-step solution for puzzle #22

What follows is a complete solve of puzzle #22, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 10 queen placements and 6 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: region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, last square in a region.

Show the full walkthrough (7 steps) — contains the solution

  1. Step 1. Sea green cannot leave row 10 — every one of its open squares (3 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 8 squares in all.

  2. Step 2. Pale yellow is down to 4 candidates in rows 7–9, columns 1 and 2, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.

  3. Step 3. The slate region has 3 squares left, spread across rows 5 and 6, columns 1 and 2. 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.

  4. Step 4. Here is the pairing argument. The red, pale yellow, khaki, and orange regions have no open squares outside rows 6–9 — 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: 6 squares come off in one stroke.

    Puzzle #22 after step 4: 0 of 10 queens placed
    The board after step 4 — 0 of 10 queens placed.

  5. Step 5. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 8 squares in all.

  6. Step 6. Here is the pairing argument. The slate and pale yellow regions have no open squares outside columns 1 and 2 — 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: 7 squares come off in one stroke.

  7. Step 7. From here the board cascades. Keep alternating the region sweep and the line sweep and 10 queens fall out in order without any new idea being needed: mint has one square left, so its queen takes row 4, column 3; indigo has one square left, so its queen takes row 1, column 4; 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 2; khaki has one square left, so its queen takes row 8, column 5; sea green has one square left, so its queen takes row 10, column 6; red has one square left, so its queen takes row 6, column 7; blue has one square left, so its queen takes row 3, column 8; amber has one square left, so its queen takes row 2, column 10; 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 #22 after step 7: 10 of 10 queens placed
    The board after step 7 — 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 #22

Board #22 of the 10x10 bank carries a easy rating. Its 100 squares are shared between ten colored regions, and the solution places ten queens: one per row, one per column, one per region, none of them side by side or corner to corner.

There is exactly one way to finish #22, 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 #22 are: sea green (3 squares), slate (4 squares), indigo (5 squares), pale yellow (6 squares), red (9 squares), khaki (10 squares), mint (12 squares), orange (15 squares), amber (16 squares), blue (20 squares). An even split would hand every color 10 squares, so the sea green region is running 7 squares under that average while the blue region is the roomiest area on the grid with 20 squares.

That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Sea green has only 3 places to put its queen; blue has 20. The spread between them on this board is 17 squares, and the wider that spread runs, the more it pays to solve from the small end inward.

Shape matters as much as size. The slate region is the most compact color here, filling 100% of the 2×2 box that encloses it, while blue is the most scattered at 48% of its 6×7 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

Here is the full footprint of every color on puzzle #22, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Sea green covers 3 squares, spanning 1 row and 3 columns (rows 10–10, columns 4–6). It never leaves row 10, so its queen is locked to that row.

Slate occupies 4 squares within rows 5–6, columns 1–2.

Indigo: 5 squares across rows 1–2, columns 1–4.

Pale yellow covers 6 squares, spanning 4 rows and 2 columns (rows 7–10, columns 1–2).

Red occupies 9 squares within rows 6–9, columns 5–8.

Khaki: 10 squares across rows 7–10, columns 2–5.

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

Orange occupies 15 squares within rows 6–10, columns 7–10.

Amber: 16 squares across rows 1–6, columns 6–10.

Blue covers 20 squares, spanning 6 rows and 7 columns (rows 1–6, columns 3–9).

Where to start on this board

The most constrained color is where a solve should begin, and on #22 that is sea green with 3 squares. It reaches from row 10 to row 10 and from column 4 to column 6 — 1 row by 3 columns. 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 sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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. Indigo and slate 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 #22

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3 and 4 are the narrowest, touched by only 3 colors. Column 10 is the tightest column, with 2 colors.

Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 6 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.

Working through the middle game

The middle of #22 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 #22 is rated easy

The rating reflects how much the board concedes up front. Puzzle #22 has one region confined to a single row or column, its smallest color holds 3 squares, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 17 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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 #22 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.

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 #22 the place to look for one is around column 10.

Use this as a rough check on your progress: the completed board has 4 queens around the border and 6 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.

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 #22?

Puzzle #22 is rated easy. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 10x10 puzzle #22 have only one solution?

Yes. Every puzzle in the 10x10 collection, including #22, 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 #22?

Start with the sea green region — the most constrained color on this board, with 3 squares spanning rows 10–10 and columns 4–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.