10x10 Queens Puzzle #151
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 17-step reasoning is written out below.
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Step-by-step solution for puzzle #151
What follows is a complete solve of puzzle #151, worked the way a person works it rather than the way a computer does. It runs to 17 deductions — 10 queen placements and 7 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 columns.
Techniques this board needs: last square in a region, region shadow, region confined to one line, line confined to one region, last square in a column, pairing argument on rows, pairing argument on columns, last square in a row.
Show the full walkthrough (11 steps) — contains the solution
Step 1. Only row 1, column 1 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 1, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. The mint region is down to one open square, at row 4, column 10. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 4 and column 10, the remaining mint squares, and the ring of neighbours around row 4, column 10.
Step 3. The slate region has 3 squares left, spread across rows 3 and 5, columns 3–5. 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 4. All of the sea green region's remaining squares (2 open squares) lie in row 10. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 8 squares come off.
The board after step 4 — 2 of 10 queens placed. Step 5. Khaki is down to 3 candidates in rows 8 and 9, columns 3 and 4, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about khaki itself.
Step 6. The pale yellow region has 4 squares left, spread across rows 6 and 7, columns 4–7. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 7. Column 8 has only red squares left (7 open squares). The column needs a queen and only red can provide one, which pins that colour to this column and kills its squares elsewhere: 9 squares.
The board after step 7 — 2 of 10 queens placed. Step 8. Column 9 now has exactly one open square, row 2, column 9, in the amber region. Columns must each hold a queen, so that placement is forced. Everything else in row 2, in column 9, in amber and in the ring around that square is now dead.
Step 9. Khaki and orange are now boxed into rows 8 and 9 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 3 squares at once.
Step 10. Here is the pairing argument. The blue, khaki, and sea green regions have no open squares outside columns 2–4 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.
Step 11. 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: slate has one square left, so its queen takes row 5, column 5; row 3 is down to one square, putting a queen on row 3, column 2; sea green has one square left, so its queen takes row 10, column 3; khaki has one square left, so its queen takes row 8, column 4; row 6 is down to one square, putting a queen on row 6, column 8; 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 7. 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 7.
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 #151
10x10 puzzle #151 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.
Like every board on this site, #151 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 #151 are: indigo (1 square), mint (1 square), sea green (4 squares), slate (5 squares), khaki (5 squares), pale yellow (6 squares), orange (13 squares), amber (14 squares), blue (25 squares), red (26 squares). An even split would hand every color 10 squares, so the indigo region is running 9 squares under that average while the red region is the roomiest area on the grid with 26 squares.
That imbalance is the first thing to read, because every region takes exactly one queen regardless of how much space it owns. Indigo has only 1 place to put its queen; red has 26. The spread between them on this board is 25 squares, and the wider that spread runs, the more it pays to solve from the small end inward.
Look at arrangement as well as area. Indigo is the densest region on this grid at 100% of a 1×1 frame; pale yellow is the loosest, occupying only 50% of its 3×4 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
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.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 1–1). It never leaves row 1, so its queen is locked to that row.
Mint occupies 1 square within rows 4–4, columns 10–10. It never leaves row 4, so its queen is locked to that row.
Sea green: 4 squares across rows 9–10, columns 1–3.
Slate covers 5 squares, spanning 3 rows and 3 columns (rows 3–5, columns 3–5).
Khaki occupies 5 squares within rows 8–9, columns 2–4.
Pale yellow: 6 squares across rows 6–8, columns 4–7.
Orange covers 13 squares, spanning 3 rows and 7 columns (rows 8–10, columns 4–10).
Amber occupies 14 squares within rows 1–3, columns 5–10.
Blue: 25 squares across rows 1–8, columns 1–5.
Red covers 26 squares, spanning 8 rows and 6 columns (rows 2–9, columns 5–10).
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 1 and 1. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #151.
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.
Work outward to the other flat regions next. Mint, khaki and sea green 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 #151
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, 8 and 9 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 8 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #151 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.
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.
The trap on a board like this is placing a queen that is merely plausible. Every placement should be traceable to an elimination you can point at. If you cannot name the rule that forced it, you are guessing — and on a single-solution grid a guess costs more to unwind than the deduction would have taken to find.
Why puzzle #151 is rated easy
The easy label comes out of four measurable things about #151: four 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 25. Boards that score tight on those measures give up their first queen fast.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
If #151 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.
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 #151 the place to look for one is around row 10.
A non-spoiling distribution check: solved, #151 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 #151?
Puzzle #151 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #151 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #151, 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 #151?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.