10x10 Queens Puzzle #159
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 21-step reasoning is written out below.
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Step-by-step solution for puzzle #159
What follows is a complete solve of puzzle #159, worked the way a person works it rather than the way a computer does. It runs to 21 deductions — 10 queen placements and 11 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, region confined to one line, pairing argument on rows, last square in a region, last square in a column, last square in a row.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Wherever blue ends up among its 4 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 2. Slate is down to 4 candidates in rows 4 and 5, columns 8–10, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 3. Red cannot leave column 10 — every one of its open squares (3 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 8 squares in all.
Step 4. Wherever slate 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 — 0 of 10 queens placed. Step 5. Sea green cannot leave row 10 — every one of its open squares (3 open squares) sits there. That row belongs to sea green, so every square on it owned by another colour is dead: 7 squares in all.
Step 6. The khaki region has 5 squares left, spread across rows 8 and 9, columns 2–5. 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 7. Orange cannot leave row 9 — every one of its open squares (4 open squares) sits there. That row belongs to orange, so every square on it owned by another colour is dead: 4 squares in all.
Step 8. All of the khaki region's remaining squares (4 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
Step 9. Mint, slate, red, and pale yellow are now boxed into rows 4–7 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 6 squares at once.
The board after step 9 — 0 of 10 queens placed. Step 10. The blue region is down to one open square, at row 3, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 1, the remaining blue squares, and the ring of neighbours around row 3, column 1.
Step 11. Only row 1, column 2 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 2, in indigo and in the ring around that square is now dead.
Step 12. All of the khaki region's remaining squares (3 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 2 squares come off.
Step 13. Only row 9, column 3 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 3, the rest of the orange region and every square adjacent to it — diagonals included.
The board after step 13 — 3 of 10 queens placed. Step 14. The khaki region is down to one open square, at row 8, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 5, the remaining khaki squares, and the ring of neighbours around row 8, column 5.
Step 15. Scan column 4 and only row 2, column 4 remains — a amber square. Every column takes a queen, so it takes this one. Cross out the whole of row 2 and column 4, the remaining amber squares, and the ring of neighbours around row 2, column 4.
Step 16. Wherever mint 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 17. 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: column 6 is down to one square, putting a queen on row 4, column 6; slate has one square left, so its queen takes row 5, column 8; row 6 is down to one square, putting a queen on row 6, column 10; pale yellow has one square left, so its queen takes row 7, column 7; sea green has one square left, so its queen takes row 10, column 9. 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 10, column 9.
The board after step 17 — 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 #159
10x10 puzzle #159 is rated easy. Ten colored regions cover its 100 squares, and your job is to seat ten queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
As with every puzzle in this collection, #159 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 #159 are: blue (4 squares), slate (4 squares), red (4 squares), sea green (4 squares), khaki (7 squares), orange (8 squares), indigo (10 squares), mint (18 squares), pale yellow (19 squares), amber (22 squares). An even split would hand every color 10 squares, so the blue region is running 6 squares under that average while the amber region is the roomiest area on the grid with 22 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Blue has only 4 places to put its queen; amber has 22. The spread between them on this board is 18 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Square count is only half the story — how a color is arranged counts too. Orange is the tidiest shape on #159, packing 100% of its 2×4 bounding box, whereas indigo straggles across its 6×4 box at just 42%. 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.
Blue: 4 squares across rows 3–4, columns 1–3.
Slate covers 4 squares, spanning 2 rows and 3 columns (rows 4–5, columns 8–10).
Red occupies 4 squares within rows 5–7, columns 9–10.
Sea green: 4 squares across rows 10–10, columns 7–10. It never leaves row 10, so its queen is locked to that row.
Khaki covers 7 squares, spanning 3 rows and 5 columns (rows 8–10, columns 2–6).
Orange occupies 8 squares within rows 9–10, columns 1–4.
Indigo: 10 squares across rows 1–6, columns 1–4.
Mint covers 18 squares, spanning 5 rows and 7 columns (rows 4–8, columns 1–7).
Pale yellow occupies 19 squares within rows 5–9, columns 6–10.
Amber: 22 squares across rows 1–3, columns 3–10.
Where to start on this board
Open on the tightest color. Here that is blue, confined to 4 squares between rows 3 and 4 and columns 1 and 3. Its queen has nowhere else to go, which means those 2 rows and 3 columns carry more constraint than anywhere else on #159.
Blue does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 4 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.
Work outward to the other flat regions next. Sea green, slate 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 #159
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 2 are the narrowest, touched by only 2 colors. Column 5 is the tightest column, 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 5 is the crowded extreme at 5 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.
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.
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 #159 is rated easy
The easy label comes out of four measurable things about #159: one of its regions is locked to one row or column, its smallest color spans 4 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 18. Boards that score tight on those measures give up their first queen fast.
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.
One pattern is worth learning at every size, and it is available here. When two regions can only fit inside the same two rows, they must take one row each — which locks every other region out of both. The same argument works for columns, and for any matching count: three regions sharing three rows, four sharing four. Spotting these pairings turns a slow grind into a quick cascade, and on #159 the place to look for one is around row 1.
A non-spoiling distribution check: solved, #159 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 #159?
Puzzle #159 is rated easy. Its smallest color region holds 4 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 #159 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #159, 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 #159?
Start with the blue region — the most constrained color on this board, with 4 squares spanning rows 3–4 and columns 1–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.