10x10 Queens Puzzle #296
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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #296
What follows is a complete solve of puzzle #296, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 10 queen placements and 12 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: region shadow, region confined to one line, pairing argument on rows, proof by contradiction, last square in a region, last square in a row.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Wherever indigo ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Blue is down to 4 candidates in rows 2 and 3, columns 8–10, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.
Step 3. Slate cannot leave row 5 — every one of its open squares (3 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 9 squares in all.
Step 4. 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: 10 squares come off in one stroke.
Step 5. Blue cannot leave row 3 — every one of its open squares (3 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 7 squares in all.
The board after step 5 — 0 of 10 queens placed. Step 6. 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: 5 squares come off.
Step 7. Test row 1, column 4 for indigo: place a queen there, run the forced consequences through, and the slate region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 6. Follow the forced moves that would create and the mint region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 9. Test row 2, column 6 for indigo: place a queen there, run the forced consequences through, and the sea green region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 9 — 0 of 10 queens placed. Step 10. 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: indigo has one square left, so its queen takes row 1, column 5; amber has one square left, so its queen takes row 2, column 10; blue has one square left, so its queen takes row 3, column 8; mint has one square left, so its queen takes row 4, column 6; slate has one square left, so its queen takes row 5, column 4. 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 4.
Step 11. The pale yellow region has 3 squares left, spread across rows 6–8, columns 3 and 7. 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 12. All of the red 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.
Step 13. Orange cannot leave column 7 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 2 squares in all.
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: pale yellow has one square left, so its queen takes row 7, column 3; red has one square left, so its queen takes row 6, column 9; row 8 is down to one square, putting a queen on row 8, column 1; 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 7. 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 7.
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 #296
This is puzzle number 296 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.
Like every board on this site, #296 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 #296 are: slate (3 squares), indigo (4 squares), amber (4 squares), blue (4 squares), sea green (5 squares), red (12 squares), pale yellow (13 squares), orange (14 squares), khaki (18 squares), mint (23 squares). An even split would hand every color 10 squares, so the slate region is running 7 squares under that average while the mint region is the roomiest area on the grid with 23 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Slate is working with 3 squares while mint has 23 to spare, a gap of 20. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 2×2 frame; mint is the loosest, occupying only 57% of its 5×8 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.
Slate occupies 3 squares within rows 5–5, columns 4–6. It never leaves row 5, so its queen is locked to that row.
Indigo: 4 squares across rows 1–2, columns 4–6.
Amber covers 4 squares, spanning 2 rows and 2 columns (rows 1–2, columns 9–10).
Blue occupies 4 squares within rows 2–3, columns 8–10.
Sea green: 5 squares across rows 9–10, columns 1–4.
Red covers 12 squares, spanning 4 rows and 4 columns (rows 4–7, columns 7–10).
Pale yellow occupies 13 squares within rows 6–9, columns 3–7.
Orange: 14 squares across rows 8–10, columns 5–10.
Khaki covers 18 squares, spanning 6 rows and 4 columns (rows 4–9, columns 1–4).
Mint occupies 23 squares within rows 1–5, columns 1–8.
Where to start on this board
Slate is the natural opening on #296: it is the smallest region on the grid at 3 squares, boxed into rows 5–5 and columns 4–6. Since one of those squares must hold a queen, that block of 1 row and 3 columns is the most contested part of the board.
Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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, amber, blue and sea green 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 #296
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3 and 10 are the narrowest, touched by only 2 colors. Columns 1 and 2 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 9 is the crowded extreme at 4 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 #296 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 #296 is rated hard
Ratings track how generous the opening is. On #296 you get one region pinned to a single row or column, a smallest color of 3 squares, a tightest row crossed by 2 colors, and a 20-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
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 #296 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.
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 3 on this board.
A non-spoiling distribution check: solved, #296 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 #296?
Puzzle #296 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #296 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #296, 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 #296?
Start with the slate region — the most constrained color on this board, with 3 squares spanning rows 5–5 and columns 4–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.