10x10 Queens Puzzle #56
Difficulty: medium. 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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #56
What follows is a complete solve of puzzle #56, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 10 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 proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, proof by contradiction, last square in a column.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Only row 4, column 1 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 1, the rest of the mint region and every square adjacent to it — diagonals included.
Step 2. The indigo region is down to one open square, at row 1, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 2, the remaining indigo squares, and the ring of neighbours around row 1, column 2.
Step 3. Only row 6, column 3 is still open in red, and since each colour must take a queen, that settles it. Everything else in row 6, in column 3, in red and in the ring around that square is now dead.
Step 4. All of the orange region's remaining squares (2 open squares) lie in column 10. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 7 squares come off.
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: 6 squares in all.
The board after step 5 — 3 of 10 queens placed. Step 6. The orange region is down to one open square, at row 9, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 10, the remaining orange squares, and the ring of neighbours around row 9, column 10.
Step 7. Wherever khaki ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. Pale yellow is down to 4 candidates in rows 7 and 8, columns 4, 6, and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. Pale yellow and khaki are now boxed into rows 7 and 8 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.
The board after step 9 — 4 of 10 queens placed. Step 10. Here is the pairing argument. The blue and khaki regions have no open squares outside columns 8 and 9 — 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.
Step 11. Only row 10, column 7 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 7, in sea green and in the ring around that square is now dead.
Step 12. The amber region has 4 squares left, spread across rows 2 and 3, columns 4–6. 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 13. Test row 2, column 5 for amber: place a queen there, run the forced consequences through, and row 7 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: column 5 is down to one square, putting a queen on row 5, column 5; column 8 is down to one square, putting a queen on row 8, column 8; pale yellow has one square left, so its queen takes row 7, column 6; 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. 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 3, column 9.
The board after step 14 — 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 #56
This is puzzle number 56 in the 10x10 collection, rated medium. 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 #56 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 #56 are: mint (1 square), orange (2 squares), sea green (3 squares), indigo (5 squares), red (7 squares), khaki (11 squares), amber (13 squares), blue (13 squares), slate (19 squares), pale yellow (26 squares). An even split would hand every color 10 squares, so the mint region is running 9 squares under that average while the pale yellow region is the roomiest area on the grid with 26 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Mint is working with 1 square while pale yellow has 26 to spare, a gap of 25. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Mint is the densest region on this grid at 100% of a 1×1 frame; slate is the loosest, occupying only 45% of its 7×6 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.
Mint occupies 1 square within rows 4–4, columns 1–1. It never leaves row 4, so its queen is locked to that row.
Orange: 2 squares across rows 9–10, columns 10–10. It sits wholly inside column 10, so its queen must be in that column.
Sea green covers 3 squares, spanning 1 row and 3 columns (rows 10–10, columns 7–9). It never leaves row 10, so its queen is locked to that row.
Indigo occupies 5 squares within rows 1–3, columns 1–2.
Red: 7 squares across rows 4–6, columns 1–3.
Khaki covers 11 squares, spanning 4 rows and 4 columns (rows 6–9, columns 7–10).
Amber occupies 13 squares within rows 1–3, columns 2–7.
Blue: 13 squares across rows 1–5, columns 8–10.
Slate covers 19 squares, spanning 7 rows and 6 columns (rows 2–8, columns 3–8).
Pale yellow occupies 26 squares within rows 6–10, columns 1–8.
Where to start on this board
Mint is the natural opening on #56: it is the smallest region on the grid at 1 square, boxed into rows 4–4 and columns 1–1. 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 mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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 #56
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 5, 7, 8, 9 and 10 are the narrowest, touched by only 3 colors. Columns 4, 5, 6, 9 and 10 are the tightest columns, with 3 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 4 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #56 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 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.
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 #56 is rated medium
Ratings track how generous the opening is. On #56 you get four regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 25-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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.
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 1 on this board.
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
- 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 #56?
Puzzle #56 is rated medium. 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 #56 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #56, 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 #56?
Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.