10x10 Queens Puzzle #29
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 14-step reasoning is written out below.
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Step-by-step solution for puzzle #29
What follows is a complete solve of puzzle #29, worked the way a person works it rather than the way a computer does. It runs to 14 deductions — 10 queen placements and 4 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: region shadow, last square in a region, region confined to one line, pairing argument on rows, last square in a column.
Show the full walkthrough (8 steps) — contains the solution
Step 1. Wherever indigo ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 2. The blue region is down to one open square, at row 3, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 9, the remaining blue squares, and the ring of neighbours around row 3, column 9.
Step 3. Only row 5, column 10 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 10, in slate and in the ring around that square is now dead.
Step 4. The indigo region is down to one open square, at row 1, column 8. Every colour holds exactly one queen, so that square is forced. Row 1 and column 8 are now spoken for, along with the squares touching that queen.
The board after step 4 — 3 of 10 queens placed. Step 5. Mint is down to 2 candidates in rows 2 and 4, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about mint itself.
Step 6. All of the khaki region's remaining squares (2 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 9 squares come off.
Step 7. Amber and mint are now boxed into rows 2 and 4 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 5 squares at once.
Step 8. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: red has one square left, so its queen takes row 6, column 1; sea green has one square left, so its queen takes row 10, column 2; khaki has one square left, so its queen takes row 8, column 3; column 4 is down to one square, putting a queen on row 2, column 4; mint has one square left, so its queen takes row 4, column 7; pale yellow has one square left, so its queen takes row 7, column 6; orange has one square left, so its queen takes row 9, column 5. 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 5.
The board after step 8 — 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 #29
Puzzle #29 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.
As with every puzzle in this collection, #29 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #29 are: blue (2 squares), khaki (2 squares), slate (4 squares), indigo (5 squares), mint (9 squares), sea green (10 squares), amber (12 squares), red (14 squares), pale yellow (16 squares), orange (26 squares). An even split would hand every color 10 squares, so the blue region is running 8 squares under that average while the orange region is the roomiest area on the grid with 26 squares.
Every color takes one queen and one only, whatever its size — which is why the 24 square difference between blue 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 blue is the most solid, filling 100% of its 2×1 box, and indigo the most diffuse at 56% of its 3×3 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #29, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Blue occupies 2 squares within rows 2–3, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.
Khaki: 2 squares across rows 8–9, columns 3–3. It sits wholly inside column 3, so its queen must be in that column.
Slate covers 4 squares, spanning 4 rows and 1 column (rows 2–5, columns 10–10). It sits wholly inside column 10, so its queen must be in that column.
Indigo occupies 5 squares within rows 1–3, columns 8–10.
Mint: 9 squares across rows 1–4, columns 6–9.
Sea green covers 10 squares, spanning 5 rows and 3 columns (rows 6–10, columns 1–3).
Amber occupies 12 squares within rows 1–4, columns 1–5.
Red: 14 squares across rows 2–6, columns 1–4.
Pale yellow covers 16 squares, spanning 4 rows and 7 columns (rows 4–7, columns 4–10).
Orange occupies 26 squares within rows 7–10, columns 3–10.
Where to start on this board
Begin where the choices are fewest. On this board that means blue, which owns just 2 squares inside a 2×1 window running from row 2 to row 3 and column 9 to column 9. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because blue stays inside column 9, its queen belongs to that column and no other region may use it. Cross out the rest of column 9 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Khaki is no more than two rows tall, so its queen is 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 #29
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, 2, 5, 6 and 7 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 3, meeting 6 colors, is the opposite case — the busiest line on #29 and the last one you should expect to crack.
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.
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.
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 #29 is rated hard
The rating reflects how much the board concedes up front. Puzzle #29 has three regions confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 24 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
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 #29 is row 10.
Here is a shape check that gives nothing away: when #29 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 #29?
Puzzle #29 is rated hard. Its smallest color region holds 2 squares, 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 #29 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #29, 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 #29?
Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 2–3 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.