10x10 Queens Puzzle #293
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 19-step reasoning is written out below.
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Step-by-step solution for puzzle #293
What follows is a complete solve of puzzle #293, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 10 queen placements and 9 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, pairing argument on rows, last square in a region, region confined to one line, line confined to one region, last square in a column.
Show the full walkthrough (15 steps) — contains the solution
Step 1. Wherever khaki ends up among its 5 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 2. Sea green is down to 3 candidates in rows 9 and 10, columns 3 and 4, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about sea green itself.
Step 3. The khaki region has 4 squares left, spread across rows 8–10, columns 1 and 2. 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. Here is the pairing argument. The khaki, orange, and sea green regions have no open squares outside rows 8–10 — 3 colours confined to 3 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. Only row 7, column 8 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 8, the rest of the pale yellow region and every square adjacent to it — diagonals included.
The board after step 5 — 1 of 10 queens placed. Step 6. The red region is down to one open square, at row 6, column 4. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 6 and column 4, the remaining red squares, and the ring of neighbours around row 6, column 4.
Step 7. Only row 10, column 3 is still open in sea green, and since each colour must take a queen, that settles it. Everything else in row 10, in column 3, in sea green and in the ring around that square is now dead.
Step 8. Indigo is down to 4 candidates in rows 1, 3, 4, and 5, columns 2 and 5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about indigo itself.
Step 9. Mint cannot leave column 9 — every one of its open squares (2 open squares) sits there. That column belongs to mint, so every square in it owned by another colour is dead: 5 squares in all.
Step 10. Wherever slate 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.
The board after step 10 — 3 of 10 queens placed. Step 11. Column 10 has only amber squares left (3 open squares). The column needs a queen and only amber can provide one, which pins that colour to this column and kills its squares elsewhere: 8 squares.
Step 12. Column 7 now has exactly one open square, row 5, column 7, in the slate region. Columns must each hold a queen, so that placement is forced. Everything else in row 5, in column 7, in slate and in the ring around that square is now dead.
Step 13. Only row 4, column 9 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 9, the rest of the mint region and every square adjacent to it — diagonals included.
Step 14. Every open square left in column 6 (2 open squares) belongs to the orange region. Since that column must contain a queen, orange is the colour that supplies it — so orange squares anywhere else on the board are out, 2 squares in all.
Step 15. 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 5 is down to one square, putting a queen on row 1, column 5; amber has one square left, so its queen takes row 2, column 10; blue has one square left, so its queen takes row 3, column 1; khaki has one square left, so its queen takes row 8, column 2; orange has one square left, so its queen takes row 9, column 6. 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 6.
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 #293
Puzzle #293 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 #293, 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 #293 are: sea green (4 squares), red (5 squares), khaki (5 squares), mint (6 squares), pale yellow (8 squares), blue (11 squares), orange (11 squares), slate (15 squares), indigo (16 squares), amber (19 squares). An even split would hand every color 10 squares, so the sea green region is running 6 squares under that average while the amber region is the roomiest area on the grid with 19 squares.
Every color takes one queen and one only, whatever its size — which is why the 15 square difference between sea green and amber is worth noting before you touch the grid. The cramped colors resolve first and hand you information; the roomy ones absorb whatever is left over at the end.
A region's outline can constrain it more than its size does. Here red is the most solid, filling 83% of its 3×2 box, and indigo the most diffuse at 40% of its 8×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Sea green occupies 4 squares within rows 9–10, columns 2–4.
Red: 5 squares across rows 6–8, columns 3–4.
Khaki covers 5 squares, spanning 3 rows and 3 columns (rows 8–10, columns 1–3).
Mint occupies 6 squares within rows 4–7, columns 9–10.
Pale yellow: 8 squares across rows 7–9, columns 7–10.
Blue covers 11 squares, spanning 6 rows and 3 columns (rows 1–6, columns 1–3).
Orange occupies 11 squares within rows 8–10, columns 5–10.
Slate: 15 squares across rows 3–7, columns 5–8.
Indigo covers 16 squares, spanning 8 rows and 5 columns (rows 1–8, columns 1–5).
Amber occupies 19 squares within rows 1–5, columns 5–10.
Where to start on this board
Begin where the choices are fewest. On this board that means sea green, which owns just 4 squares inside a 2×3 window running from row 9 to row 10 and column 2 to column 4. A queen is guaranteed somewhere in there, so every line that window crosses is under pressure from the very first move.
With 2 rows and 3 columns of room, sea green needs narrowing rather than solving outright. Test its 4 squares against the neighbours: a candidate that would strand another color is dead, and eliminating even two of them tightens the whole corner of the board around rows 9–10.
No region here is pinned to a shallow band of rows, so turn the same idea sideways: mint and red span only two columns. Two narrow regions sharing the same two columns take one column each and lock everyone else out of both, which is usually enough to crack a board that offers no free opening.
The tightest lines on puzzle #293
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2 and 10 are the narrowest, touched by only 3 colors. Columns 1, 4 and 6 are the tightest columns, with 3 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 9 rows. Row 8 sits at the far end with 5 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #293 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 #293 is rated easy
The easy label comes out of four measurable things about #293: it has no region locked to one row or column, its smallest color spans 4 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 15. 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.
Stuck on #293 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.
There is one technique that repays study at every grid size. Two regions restricted to the same two rows must occupy one row apiece, which excludes all other regions from those rows entirely; the identical logic applies to columns and to larger matched sets. It converts slow boards into fast ones, and the likeliest place to find it on #293 is row 1.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 6 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #293?
Puzzle #293 is rated easy. Its smallest color region holds 4 squares, its tightest row is touched by 3 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 10x10 puzzle #293 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #293, 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 #293?
Start with the sea green region — the most constrained color on this board, with 4 squares spanning rows 9–10 and columns 2–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.