10x10 Queens Puzzle #126
Difficulty: easy. 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 #126
What follows is a complete solve of puzzle #126, 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 shadow, region confined to one line, pairing argument on rows.
Show the full walkthrough (12 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. The orange region is down to one open square, at row 9, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 8, the remaining orange squares, and the ring of neighbours around row 9, column 8.
Step 3. Only row 3, column 7 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 7, in blue and in the ring around that square is now dead.
Step 4. Wherever slate 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.
The board after step 4 — 3 of 10 queens placed. Step 5. Red is down to 5 candidates in rows 6–8, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 6. The pale yellow region has 2 squares left, spread across rows 7 and 8, columns 2 and 5. You do not need to know which of them takes the queen: every one of those options attacks the same 5 squares, so they can be crossed out now.
Step 7. Wherever red ends up among its 2 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 8. All of the khaki 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: 5 squares come off.
The board after step 8 — 3 of 10 queens placed. Step 9. Slate cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 3 squares in all.
Step 10. Wherever indigo 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.
Step 11. 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 2 squares at once.
Step 12. 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 4; slate has one square left, so its queen takes row 5, column 9; mint has one square left, so its queen takes row 4, column 5; indigo has one square left, so its queen takes row 1, column 6; pale yellow has one square left, so its queen takes row 7, column 2; amber has one square left, so its queen takes row 2, column 3; khaki has one square left, so its queen takes row 8, column 10. 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 8, column 10.
The board after step 12 — 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 #126
Board #126 of the 10x10 bank carries a easy rating. Its 100 squares are shared between ten colored regions, and the solution places ten queens: one per row, one per column, one per region, none of them side by side or corner to corner.
Like every board on this site, #126 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 #126 are: sea green (1 square), blue (2 squares), khaki (5 squares), red (6 squares), orange (6 squares), slate (7 squares), indigo (13 squares), pale yellow (15 squares), amber (20 squares), mint (25 squares). An even split would hand every color 10 squares, so the sea green region is running 9 squares under that average while the mint region is the roomiest area on the grid with 25 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 mint with 25. A 24-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 #126, packing 100% of its 1×2 bounding box, whereas slate straggles across its 5×3 box at just 47%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a easy board.
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 1 row and 2 columns (rows 3–3, columns 7–8). It never leaves row 3, so its queen is locked to that row.
Khaki occupies 5 squares within rows 6–9, columns 9–10.
Red: 6 squares across rows 6–9, columns 2–4.
Orange covers 6 squares, spanning 2 rows and 5 columns (rows 9–10, columns 6–10).
Slate occupies 7 squares within rows 2–6, columns 8–10.
Indigo: 13 squares across rows 1–3, columns 4–10.
Pale yellow covers 15 squares, spanning 5 rows and 6 columns (rows 6–10, columns 1–6).
Amber occupies 20 squares within rows 1–6, columns 1–4.
Mint: 25 squares across rows 2–9, columns 4–9.
Where to start on this board
The most constrained color is where a solve should begin, and on #126 that is sea green with 1 square. It reaches from row 10 to row 10 and from column 1 to column 1 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
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 #126
A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 2 colors. Columns 1, 2, 3 and 5 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 6 is the crowded extreme at 6 colors — worth revisiting later, not now.
Working through the middle game
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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 #126 is rated easy
The easy label comes out of four measurable things about #126: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 24. Boards that score tight on those measures give up their first queen fast.
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 easy board from the rest.
If #126 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.
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 #126, start hunting for it near row 1.
A non-spoiling distribution check: solved, #126 carries 3 queens on the perimeter against 7 queens inside, 1 of them cornered. 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 #126?
Puzzle #126 is rated easy. 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 #126 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #126, 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 #126?
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.