10x10 Queens Puzzle #176
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 20-step reasoning is written out below.
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Step-by-step solution for puzzle #176
What follows is a complete solve of puzzle #176, worked the way a person works it rather than the way a computer does. It runs to 20 deductions — 10 queen placements and 10 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, last square in a region, pairing argument on rows, region confined to one line, line confined to one region, last square in a column.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Wherever slate ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Khaki is down to 4 candidates in rows 8–10, columns 6 and 7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about khaki itself.
Step 3. Only row 10, column 5 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 5, in sea green and in the ring around that square is now dead.
Step 4. Wherever amber 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 5. Khaki is down to 2 candidates in rows 8 and 9, columns 6 and 7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about khaki itself.
The board after step 5 — 1 of 10 queens placed. Step 6. The orange region has 5 squares left, spread across rows 6, 7, and 9, columns 8 and 9. 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. 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 7 squares at once.
Step 8. All of the blue region's remaining squares (3 open squares) lie in row 3. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
Step 9. Only row 4, column 10 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 10, the rest of the mint region and every square adjacent to it — diagonals included.
The board after step 9 — 2 of 10 queens placed. Step 10. The slate region is down to one open square, at row 5, column 1. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 1, the remaining slate squares, and the ring of neighbours around row 5, column 1.
Step 11. Indigo is down to 4 candidates in rows 1 and 2, columns 2–4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 12. All of the red region's remaining squares (2 open squares) lie in row 6. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
Step 13. Column 9 has only orange squares left (2 open squares). The column needs a queen and only orange can provide one, which pins that colour to this column and kills its squares elsewhere: 1 square.
Step 14. 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: column 8 is down to one square, putting a queen on row 6, column 8; orange has one square left, so its queen takes row 9, column 9; khaki has one square left, so its queen takes row 8, column 6; amber has one square left, so its queen takes row 2, column 4; indigo has one square left, so its queen takes row 1, column 2; blue has one square left, so its queen takes row 3, column 7; pale yellow has one square left, so its queen takes row 7, 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 7, column 3.
The board after step 14 — 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 #176
This is puzzle number 176 in the 10x10 collection, rated hard. The grid holds 100 squares split between ten colored regions, and a finished board carries ten queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #176, 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 #176 are: sea green (2 squares), mint (4 squares), slate (4 squares), khaki (4 squares), amber (6 squares), indigo (7 squares), orange (10 squares), blue (11 squares), red (20 squares), pale yellow (32 squares). An even split would hand every color 10 squares, so the sea green region is running 8 squares under that average while the pale yellow region is the roomiest area on the grid with 32 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Sea green is working with 2 squares while pale yellow has 32 to spare, a gap of 30. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Sea green is the densest region on this grid at 100% of a 1×2 frame; red is the loosest, occupying only 48% of its 7×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Below is every color on #176 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.
Sea green occupies 2 squares within rows 10–10, columns 5–6. It never leaves row 10, so its queen is locked to that row.
Mint: 4 squares across rows 2–4, columns 9–10.
Slate covers 4 squares, spanning 3 rows and 2 columns (rows 3–5, columns 1–2).
Khaki occupies 4 squares within rows 8–10, columns 6–7.
Amber: 6 squares across rows 1–2, columns 3–6.
Indigo covers 7 squares, spanning 3 rows and 4 columns (rows 1–3, columns 1–4).
Orange occupies 10 squares within rows 6–10, columns 7–10.
Blue: 11 squares across rows 1–3, columns 4–10.
Red covers 20 squares, spanning 7 rows and 6 columns (rows 3–9, columns 5–10).
Pale yellow occupies 32 squares within rows 3–10, columns 1–6.
Where to start on this board
Sea green is the natural opening on #176: it is the smallest region on the grid at 2 squares, boxed into rows 10–10 and columns 5–6. Since one of those squares must hold a queen, that block of 1 row and 2 columns is the most contested part of 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. Amber 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 #176
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 5, 6 and 7 are the narrowest, touched by only 3 colors. Columns 1, 2, 3 and 8 are the tightest columns, with 3 colors.
Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 9 rows. By contrast row 3 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
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.
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 #176 is rated hard
What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 3 colors, and a region-size spread of 30. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
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 hard board from the rest.
Stuck on #176 despite the hard 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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.
As a sanity check rather than a spoiler: the finished solution to #176 puts 4 queens on the outer border and 6 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.
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 #176?
Puzzle #176 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 10x10 puzzle #176 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #176, 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 #176?
Start with the sea green region — the most constrained color on this board, with 2 squares spanning rows 10–10 and columns 5–6. Narrowing the smallest region first removes the most squares elsewhere on the grid.