10x10 Queens Puzzle #224
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #224
What follows is a complete solve of puzzle #224, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 10 queen placements and 9 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 (11 steps) — contains the solution
Step 1. Only row 6, column 8 is still open in red, and since each colour must take a queen, that settles it. That clears row 6, column 8, the rest of the red region and every square adjacent to it — diagonals included.
Step 2. Indigo is down to 4 candidates in rows 1–3, columns 4 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 3. The amber region has 6 squares left, spread across rows 1–3, columns 1–3. 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.
Step 4. Wherever orange ends up among its 5 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
The board after step 4 — 1 of 10 queens placed. Step 5. Sea green cannot leave row 10 — every one of its open squares (2 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.
Step 6. All of the orange region's remaining squares (2 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
Step 7. Indigo, amber, and blue are now boxed into rows 1–3 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 8 squares at once.
The board after step 7 — 1 of 10 queens placed. Step 8. All of the mint region's remaining squares (3 open squares) lie in row 4. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 6 squares come off.
Step 9. Slate cannot leave row 5 — every one of its open squares (4 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 3 squares in all.
Step 10. Here is the pairing argument. The amber, slate, orange, and sea green regions have no open squares outside columns 1–4 — 4 colours confined to 4 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 5; blue has one square left, so its queen takes row 3, column 6; amber has one square left, so its queen takes row 2, column 1; khaki has one square left, so its queen takes row 8, column 7; pale yellow has one square left, so its queen takes row 7, column 10; mint has one square left, so its queen takes row 4, column 9; sea green has one square left, so its queen takes row 10, column 2; orange has one square left, so its queen takes row 9, column 4; slate has one square left, so its queen takes row 5, 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 5, column 3.
The board after step 11 — 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 #224
This is puzzle number 224 in the 10x10 collection, rated hard. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Puzzle #224 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #224 are: red (1 square), sea green (3 squares), indigo (4 squares), orange (5 squares), amber (6 squares), pale yellow (10 squares), blue (11 squares), mint (13 squares), slate (16 squares), khaki (31 squares). An even split would hand every color 10 squares, so the red region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 31 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Red is working with 1 square while khaki has 31 to spare, a gap of 30. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Red is the densest region on this grid at 100% of a 1×1 frame; khaki is the loosest, occupying only 49% of its 7×9 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Red occupies 1 square within rows 6–6, columns 8–8. It never leaves row 6, so its queen is locked to that row.
Sea green: 3 squares across rows 10–10, columns 1–3. It never leaves row 10, so its queen is locked to that row.
Indigo covers 4 squares, spanning 3 rows and 2 columns (rows 1–3, columns 4–5).
Orange occupies 5 squares within rows 9–10, columns 3–6.
Amber: 6 squares across rows 1–3, columns 1–3.
Pale yellow covers 10 squares, spanning 6 rows and 2 columns (rows 5–10, columns 9–10).
Blue occupies 11 squares within rows 1–3, columns 5–10.
Mint: 13 squares across rows 2–5, columns 7–10.
Slate covers 16 squares, spanning 5 rows and 5 columns (rows 2–6, columns 1–5).
Khaki occupies 31 squares within rows 4–10, columns 1–9.
Where to start on this board
Red is the natural opening on #224: it is the smallest region on the grid at 1 square, boxed into rows 6–6 and columns 8–8. Since one of those squares must hold a queen, that block of 1 row and 1 column is the most contested part of the board.
Because red never leaves row 6, its queen must be in that row — and that instantly settles row 6 for everyone else. Sweep along it and X out every square that is not red. 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. Sea green and orange 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 #224
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 7 and 8 are the narrowest, touched by only 2 colors. Columns 6, 7 and 10 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 2 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
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
Two checks, run alternately, will carry you through the 100 squares of this grid. First: has any of the ten regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 10x10 boards almost always run one of those checks and forget the other.
Resist the plausible-looking queen. On #224 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 #224 is rated hard
Ratings track how generous the opening is. On #224 you get three regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 2 colors, and a 30-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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.
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 7 on this board.
A non-spoiling distribution check: solved, #224 carries 4 queens on the perimeter against 6 queens inside, with the corners empty. 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
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- 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.
- 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 #224?
Puzzle #224 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #224 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #224, 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 #224?
Start with the red region — the most constrained color on this board, with 1 square spanning rows 6–6 and columns 8–8. Narrowing the smallest region first removes the most squares elsewhere on the grid.