10x10 Queens Puzzle #184
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 27-step reasoning is written out below.
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Step-by-step solution for puzzle #184
What follows is a complete solve of puzzle #184, worked the way a person works it rather than the way a computer does. It runs to 27 deductions — 10 queen placements and 17 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 proof by contradiction.
Techniques this board needs: region shadow, last square in a region, region confined to one line, pairing argument on rows, pairing argument on columns, proof by contradiction, last square in a column, last square in a row.
Show the full walkthrough (22 steps) — contains the solution
Step 1. Wherever indigo 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. 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 3. 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 4. All of the amber region's remaining squares (3 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.
Step 5. Slate is down to 4 candidates in rows 5, 6, and 8, columns 3 and 4, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
Step 6. The khaki region has 4 squares left, spread across rows 8 and 9, columns 6–8. 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.
The board after step 6 — 2 of 10 queens placed. Step 7. Wherever orange ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. Khaki is down to 3 candidates in rows 8 and 9, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 9. Khaki, orange, and sea green are now boxed into rows 8–10 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 7 squares at once.
Step 10. 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.
Step 11. Pale yellow cannot leave row 7 — every one of its open squares (2 open squares) sits there. That row belongs to pale yellow, so every square on it owned by another colour is dead: 7 squares in all.
The board after step 11 — 2 of 10 queens placed. Step 12. The red region has 4 squares left, spread across rows 4–6, 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 13. Amber, pale yellow, and orange are now boxed into columns 8–10 and nothing else. That is 3 colours needing 3 columns, so they fill those columns between them — which locks every remaining colour out and removes 7 squares at once.
Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 9. Follow the forced moves that would create and row 5 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Test row 2, column 10 for amber: place a queen there, run the forced consequences through, and row 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 16. The amber region is down to one open square, at row 2, column 8. Every colour holds exactly one queen, so that square is forced. Row 2 and column 8 are now spoken for, along with the squares touching that queen.
Step 17. Scan column 10 and only row 7, column 10 remains — a pale yellow square. Every column takes a queen, so it takes this one. Row 7 and column 10 are now spoken for, along with the squares touching that queen.
The board after step 17 — 4 of 10 queens placed. Step 18. Now a short proof by contradiction. Suppose the mint queen went to row 4, column 3. Follow the forced moves that would create and the red region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 19. Test row 4, column 4 for mint: place a queen there, run the forced consequences through, and row 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 20. Here is the pairing argument. The mint and khaki regions have no open squares outside columns 6 and 7 — 2 colours confined to 2 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 2 squares come off in one stroke.
Step 21. Red cannot leave column 5 — 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: 3 squares in all.
Step 22. From here the board cascades. Keep alternating the region sweep and the line sweep and 6 queens fall out in order without any new idea being needed: row 8 is down to one square, putting a queen on row 8, column 7; mint has one square left, so its queen takes row 4, column 6; red has one square left, so its queen takes row 6, column 5; slate has one square left, so its queen takes row 5, column 3; sea green has one square left, so its queen takes row 10, column 4; orange has one square left, so its queen takes row 9, 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 9, column 9.
The board after step 22 — 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 #184
This is puzzle number 184 in the 10x10 collection, rated easy. 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 #184, 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 #184 are: blue (2 squares), indigo (4 squares), khaki (4 squares), orange (4 squares), pale yellow (6 squares), amber (9 squares), red (10 squares), sea green (12 squares), slate (14 squares), mint (35 squares). An even split would hand every color 10 squares, so the blue region is running 8 squares under that average while the mint region is the roomiest area on the grid with 35 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Blue is working with 2 squares while mint has 35 to spare, a gap of 33. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Blue is the densest region on this grid at 100% of a 2×1 frame; red is the loosest, occupying only 42% of its 6×4 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Here is the full footprint of every color on puzzle #184, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Blue covers 2 squares, spanning 2 rows and 1 column (rows 2–3, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.
Indigo occupies 4 squares within rows 1–3, columns 1–2.
Khaki: 4 squares across rows 8–9, columns 6–8.
Orange covers 4 squares, spanning 2 rows and 3 columns (rows 9–10, columns 7–9).
Pale yellow occupies 6 squares within rows 7–10, columns 9–10.
Amber: 9 squares across rows 1–3, columns 7–10.
Red covers 10 squares, spanning 6 rows and 4 columns (rows 4–9, columns 3–6).
Sea green occupies 12 squares within rows 8–10, columns 1–6.
Slate: 14 squares across rows 4–8, columns 1–4.
Mint covers 35 squares, spanning 8 rows and 8 columns (rows 1–8, columns 3–10).
Where to start on this board
Blue is the natural opening on #184: it is the smallest region on the grid at 2 squares, boxed into rows 2–3 and columns 1–1. Since one of those squares must hold a queen, that block of 2 rows and 1 column is the most contested part of the board.
Because blue stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.
Work outward to the other flat regions next. Khaki 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 #184
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 4, 5, 6 and 10 are the narrowest, touched by only 3 colors. Columns 2, 5 and 10 are the tightest columns, with 3 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 8 is the crowded extreme at 6 colors — worth revisiting later, not now.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
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 #184 is rated easy
The rating reflects how much the board concedes up front. Puzzle #184 has one region confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 33 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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.
Stuck on #184 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.
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.
A non-spoiling distribution check: solved, #184 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 #184?
Puzzle #184 is rated easy. 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 #184 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #184, 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 #184?
Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 2–3 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.