10x10 Queens Puzzle #45
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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #45
What follows is a complete solve of puzzle #45, 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 rows.
Techniques this board needs: last square in a region, region shadow, pairing argument on rows, region confined to one line.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 5, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. Blue is down to 5 candidates in rows 2 and 3, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about blue itself.
Step 3. The pale yellow region has 7 squares left, spread across rows 3–7, columns 8–10. 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 amber ends up among its 3 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
The board after step 4 — 1 of 10 queens placed. Step 5. 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 6 squares at once.
Step 6. Here is the pairing argument. The mint and slate regions have no open squares outside rows 4 and 5 — 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: 7 squares come off in one stroke.
Step 7. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 1; pale yellow has one square left, so its queen takes row 7, column 10; amber has one square left, so its queen takes row 2, column 7; blue has one square left, so its queen takes row 3, column 3; slate has one square left, so its queen takes row 5, column 4; mint has one square left, so its queen takes row 4, column 6. 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 4, column 6.
The board after step 7 — 7 of 10 queens placed. Step 8. Khaki cannot leave column 8 — every one of its open squares (2 open squares) sits there. That column belongs to khaki, so every square in it owned by another colour is dead: 2 squares in all.
Step 9. The sea green region is down to one open square, at row 10, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 9, the remaining sea green squares, and the ring of neighbours around row 10, column 9.
Step 10. Only row 8, column 8 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 8, in khaki and in the ring around that square is now dead.
Step 11. The orange region is down to one open square, at row 9, column 2. Every colour holds exactly one queen, so that square is forced. Row 9 and column 2 are now spoken for, along with the squares touching that queen.
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 #45
Puzzle #45 is one of the hard-rated boards in the 10x10 set. Ten colors divide its 100 squares, and you win by placing ten queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
There is exactly one way to finish #45, 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 #45 are: indigo (1 square), red (4 squares), pale yellow (7 squares), amber (8 squares), sea green (9 squares), blue (10 squares), mint (10 squares), slate (13 squares), khaki (16 squares), orange (22 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the orange region is the roomiest area on the grid with 22 squares.
Every color takes one queen and one only, whatever its size — which is why the 21 square difference between indigo and orange is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 1×1 box, and pale yellow the most diffuse at 47% of its 5×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #45 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Indigo: 1 square across rows 1–1, columns 5–5. It never leaves row 1, so its queen is locked to that row.
Red covers 4 squares, spanning 3 rows and 2 columns (rows 4–6, columns 1–2).
Pale yellow occupies 7 squares within rows 3–7, columns 8–10.
Amber: 8 squares across rows 1–4, columns 7–10.
Sea green covers 9 squares, spanning 3 rows and 6 columns (rows 8–10, columns 5–10).
Blue occupies 10 squares within rows 1–3, columns 1–4.
Mint: 10 squares across rows 2–5, columns 5–9.
Slate covers 13 squares, spanning 5 rows and 5 columns (rows 1–5, columns 2–6).
Khaki occupies 16 squares within rows 6–9, columns 5–9.
Orange: 22 squares across rows 6–10, columns 1–5.
Where to start on this board
Begin where the choices are fewest. On this board that means indigo, which owns just 1 square inside a 1×1 window running from row 1 to row 1 and column 5 to column 5. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
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.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: red spans only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #45
A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 2 colors. Columns 1, 3, 4 and 10 are the tightest columns, with 3 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 9 rows. Row 4, meeting 5 colors, is the opposite case — the busiest line on #45 and the last one you should expect to crack.
Working through the middle game
The middle of #45 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.
The whole middle game reduces to two sweeps. Sweep the ten colors looking for one with a single square left; sweep the ten rows and ten columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 10x10 solve from a stalled one.
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 #45 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 1-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 21. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
Stuck on #45 despite the hard 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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #45 is row 10.
Here is a shape check that gives nothing away: when #45 is solved, 4 queens sit on the outer edge of the grid and 6 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #45?
Puzzle #45 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #45 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #45, 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 #45?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.