10x10 Queens Puzzle #177
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 #177
What follows is a complete solve of puzzle #177, 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, last square in a row.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Only row 7, column 1 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 1, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 2. Slate is down to 4 candidates in rows 5 and 6, columns 8–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
Step 3. Sea green cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to sea green, so every square in it owned by another colour is dead: 7 squares in all.
Step 4. Here is the pairing argument. The orange and sea green regions have no open squares outside rows 9 and 10 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 9 squares come off in one stroke.
Step 5. Khaki is down to 5 candidates in rows 6 and 8, columns 6–10, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
The board after step 5 — 1 of 10 queens placed. Step 6. Here is the pairing argument. The red and khaki regions have no open squares outside rows 6 and 8 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 6 squares come off in one stroke.
Step 7. 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 8. The mint region is down to one open square, at row 4, column 4. Every colour holds exactly one queen, so that square is forced. Row 4 and column 4 are now spoken for, along with the squares touching that queen.
Step 9. The orange region has 3 squares left, spread across rows 9 and 10, columns 5 and 6. 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 10. All of the red 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: 2 squares come off.
The board after step 10 — 3 of 10 queens placed. Step 11. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 5; orange has one square left, so its queen takes row 9, column 6; sea green has one square left, so its queen takes row 10, column 2; row 6 is down to one square, putting a queen on row 6, column 3. 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 3.
Step 12. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 3 squares in all.
Step 13. The blue region is down to one open square, at row 3, column 7. Every colour holds exactly one queen, so that square is forced. Row 3 and column 7 are now spoken for, along with the squares touching that queen.
Step 14. Only row 2, column 9 is still open in amber, and since each colour must take a queen, that settles it. That clears row 2, column 9, the rest of the amber region and every square adjacent to it — diagonals included.
Step 15. The khaki region is down to one open square, at row 8, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 8 and column 8, the remaining khaki squares, and the ring of neighbours around row 8, column 8.
The board after step 15 — 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 #177
Puzzle #177 is one of the easy-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.
There is exactly one way to finish #177, and it has been verified. More importantly, you can reach it without ever gambling: at every stage some square is provably right or provably wrong. Boards differ only in how deep you must dig before that proof appears, and the geometry of these ten regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #177 are: pale yellow (1 square), slate (5 squares), orange (6 squares), sea green (6 squares), amber (8 squares), mint (10 squares), red (11 squares), indigo (14 squares), khaki (19 squares), blue (20 squares). An even split would hand every color 10 squares, so the pale yellow region is running 9 squares under that average while the blue region is the roomiest area on the grid with 20 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 pale yellow, with 1 square, is under far more pressure than blue with 20. A 19-square gap like this one rewards players who start where the choices are fewest.
A region's outline can constrain it more than its size does. Here pale yellow is the most solid, filling 100% of its 1×1 box, and red the most diffuse at 46% of its 4×6 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #177 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.
Pale yellow: 1 square across rows 7–7, columns 1–1. It never leaves row 7, so its queen is locked to that row.
Slate covers 5 squares, spanning 3 rows and 3 columns (rows 5–7, columns 8–10).
Orange occupies 6 squares within rows 9–10, columns 3–6.
Sea green: 6 squares across rows 8–10, columns 1–2.
Amber covers 8 squares, spanning 4 rows and 3 columns (rows 1–4, columns 8–10).
Mint occupies 10 squares within rows 4–6, columns 1–5.
Red: 11 squares across rows 6–9, columns 1–6.
Indigo covers 14 squares, spanning 4 rows and 5 columns (rows 1–4, columns 1–5).
Khaki occupies 19 squares within rows 6–10, columns 5–10.
Blue: 20 squares across rows 1–6, columns 5–9.
Where to start on this board
Begin where the choices are fewest. On this board that means pale yellow, which owns just 1 square inside a 1×1 window running from row 7 to row 7 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 pale yellow never leaves row 7, its queen must be in that row — and that instantly settles row 7 for everyone else. Sweep along it and X out every square that is not pale yellow. 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. Orange 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 #177
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 3, 5, 8 and 10 are the narrowest, touched by only 3 colors. Column 7 is the tightest column, with 2 colors.
A line with few colors is a line close to resolving. Where only 3 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 5 colors — worth revisiting later, not now.
Working through the middle game
The middle of #177 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.
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.
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 #177 is rated easy
What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 1-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 19. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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.
Stuck on #177 despite the easy rating? The odds strongly favour a bookkeeping error over a hard chain of logic. Re-check each placed queen for its complete set of eliminations, paying particular attention to the four diagonal squares — those are the ones that get overlooked, and one of them is usually sitting on the answer.
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 #177, start hunting for it near column 7.
A non-spoiling distribution check: solved, #177 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 #177?
Puzzle #177 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #177 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #177, 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 #177?
Start with the pale yellow region — the most constrained color on this board, with 1 square spanning rows 7–7 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.