10x10 Queens Puzzle #129
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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #129
What follows is a complete solve of puzzle #129, 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 pairing argument on rows.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Only row 10, column 1 is still open in sea green, and since each colour must take a queen, that settles it. That clears row 10, column 1, the rest of the sea green region and every square adjacent to it — diagonals included.
Step 2. All of the blue 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: 11 squares come off.
Step 3. Indigo cannot leave column 10 — every one of its open squares (2 open squares) sits there. That column belongs to indigo, so every square in it owned by another colour is dead: 5 squares in all.
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. Mint is down to 3 candidates in rows 2 and 4, columns 2 and 3, and they overlap enough to cover 3 squares elsewhere. Cross those out without deciding anything about mint itself.
Step 6. All of the orange region's remaining squares (4 open squares) lie in row 9. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 7. Indigo and amber are now boxed into rows 1 and 2 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 10 squares at once.
Step 8. All of the mint region's remaining squares (2 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. 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: blue has one square left, so its queen takes row 3, column 9; indigo has one square left, so its queen takes row 1, column 10; amber has one square left, so its queen takes row 2, column 7; 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 2; red has one square left, so its queen takes row 6, column 8. 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 6, column 8.
The board after step 9 — 7 of 10 queens placed. Step 10. Pale yellow cannot leave column 6 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 3 squares in all.
Step 11. The orange region is down to one open square, at row 9, column 5. Every colour holds exactly one queen, so that square is forced. Row 9 and column 5 are now spoken for, along with the squares touching that queen.
Step 12. Only row 7, column 6 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 6, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 13. The khaki region is down to one open square, at row 8, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 3, the remaining khaki squares, and the ring of neighbours around row 8, column 3.
The board after step 13 — 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 #129
Puzzle #129 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, #129 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 #129 are: sea green (1 square), blue (2 squares), indigo (5 squares), amber (6 squares), mint (8 squares), pale yellow (10 squares), orange (10 squares), red (16 squares), slate (19 squares), khaki (23 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the khaki region is the roomiest area on the grid with 23 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So sea green, with 1 square, is under far more pressure than khaki with 23. A 22-square gap like this one rewards players who start where the choices are fewest.
Square count is only half the story — how a color is arranged counts too. Blue is the tidiest shape on #129, packing 100% of its 2×1 bounding box, whereas amber straggles across its 4×3 box at just 50%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Sea green: 1 square across rows 10–10, columns 1–1. It never leaves row 10, so its queen is locked to that row.
Blue covers 2 squares, spanning 2 rows and 1 column (rows 3–4, columns 9–9). It sits wholly inside column 9, so its queen must be in that column.
Indigo occupies 5 squares within rows 1–4, columns 9–10.
Amber: 6 squares across rows 1–4, columns 7–9.
Mint covers 8 squares, spanning 5 rows and 3 columns (rows 1–5, columns 1–3).
Pale yellow occupies 10 squares within rows 7–9, columns 6–10.
Orange: 10 squares across rows 9–10, columns 5–10.
Red covers 16 squares, spanning 5 rows and 6 columns (rows 3–7, columns 5–10).
Slate occupies 19 squares within rows 1–5, columns 2–7.
Khaki: 23 squares across rows 6–10, columns 1–6.
Where to start on this board
Begin where the choices are fewest. On this board that means sea green, which owns just 1 square inside a 1×1 window running from row 10 to row 10 and column 1 to column 1. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
Because sea green never leaves row 10, its queen must be in that row — and that instantly settles row 10 for everyone else. Sweep along it and X out every square that is not sea green. 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. Blue 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 #129
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 6 and 8 are the narrowest, touched by only 2 colors. Column 4 is the tightest column, with 2 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 2 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 4 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #129 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.
With ten queens to seat and ten regions competing over 100 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 10x10 boards quickly and players who stall halfway.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #129 is rated hard
Ratings track how generous the opening is. On #129 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 22-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
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.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #129, start hunting for it near row 6.
Use this as a rough check on your progress: the completed board has 2 queens around the border and 8 queens in the middle, including 2 in 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 #129?
Puzzle #129 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 #129 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #129, 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 #129?
Start with the sea green region — the most constrained color on this board, with 1 square spanning rows 10–10 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.