10x10 Queens Puzzle #121
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 22-step reasoning is written out below.
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Step-by-step solution for puzzle #121
What follows is a complete solve of puzzle #121, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 10 queen placements and 12 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: last square in a region, region shadow, pairing argument on rows, pairing argument on columns, line confined to one region, proof by contradiction, region confined to one line.
Show the full walkthrough (16 steps) — contains the solution
Step 1. Only row 5, column 1 is still open in slate, and since each colour must take a queen, that settles it. That clears row 5, column 1, the rest of the slate region and every square adjacent to it — diagonals included.
Step 2. Sea green is down to 5 candidates in rows 8–10, columns 5 and 6, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.
Step 3. Indigo, amber, blue, and mint are now boxed into rows 1–4 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 6 squares at once.
Step 4. The khaki region is down to one open square, at row 8, column 2. Every colour holds exactly one queen, so that square is forced. Row 8 and column 2 are now spoken for, along with the squares touching that queen.
Step 5. Amber is down to 6 candidates in rows 1–3, columns 3–5, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
The board after step 5 — 2 of 10 queens placed. Step 6. The orange region has 3 squares left, spread across rows 9 and 10, columns 3 and 4. 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. Wherever sea green ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 8. 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: 7 squares come off in one stroke.
Step 9. The pale yellow region has 3 squares left, spread across rows 6 and 7, columns 9 and 10. 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. Here is the pairing argument. The mint and pale yellow regions have no open squares outside columns 9 and 10 — 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: 3 squares come off in one stroke.
Step 11. Only row 1, column 8 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 8, in indigo and in the ring around that square is now dead.
The board after step 11 — 3 of 10 queens placed. Step 12. The amber region has 3 squares left, spread across rows 2 and 3, columns 3 and 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 13. Column 7 has only red squares left (2 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: 7 squares.
Step 14. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 5. Follow the forced moves that would create and column 3 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Amber cannot leave column 3 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 1 square in all.
Step 16. 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: orange has one square left, so its queen takes row 9, column 4; sea green has one square left, so its queen takes row 10, column 6; blue has one square left, so its queen takes row 3, column 5; amber has one square left, so its queen takes row 2, column 3; mint has one square left, so its queen takes row 4, column 10; pale yellow has one square left, so its queen takes row 7, column 9; red has one square left, so its queen takes row 6, 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 6, column 7.
The board after step 16 — 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 #121
Puzzle #121 is one of the hard-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 #121, 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 #121 are: slate (1 square), indigo (4 squares), mint (4 squares), sea green (5 squares), orange (7 squares), pale yellow (8 squares), amber (10 squares), blue (13 squares), khaki (13 squares), red (35 squares). An even split would hand every color 10 squares, so the slate region is running 9 squares under that average while the red region is the roomiest area on the grid with 35 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 slate, with 1 square, is under far more pressure than red with 35. A 34-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 slate is the most solid, filling 100% of its 1×1 box, and red the most diffuse at 49% of its 8×9 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Below is every color on #121 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.
Slate covers 1 square, spanning 1 row and 1 column (rows 5–5, columns 1–1). It never leaves row 5, so its queen is locked to that row.
Indigo occupies 4 squares within rows 1–2, columns 8–10.
Mint: 4 squares across rows 2–4, columns 9–10.
Sea green covers 5 squares, spanning 3 rows and 2 columns (rows 8–10, columns 5–6).
Orange occupies 7 squares within rows 8–10, columns 2–4.
Pale yellow: 8 squares across rows 4–9, columns 9–10.
Amber covers 10 squares, spanning 3 rows and 5 columns (rows 1–3, columns 1–5).
Blue occupies 13 squares within rows 1–4, columns 4–8.
Khaki: 13 squares across rows 3–10, columns 1–3.
Red covers 35 squares, spanning 8 rows and 9 columns (rows 3–10, columns 2–10).
Where to start on this board
Begin where the choices are fewest. On this board that means slate, which owns just 1 square inside a 1×1 window running from row 5 to row 5 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 slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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. Indigo 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 #121
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 6 and 7 are the narrowest, touched by only 3 colors. Column 7 is the tightest column, with 2 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 9 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
After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.
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 #121 is rated hard
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 34. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.
Rating is a measure of chain length, not of chance. With one guaranteed solution and no guessing required, the only thing that changes across the collection is how many eliminations you must link together before a queen is provably forced.
Stuck on #121 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.
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 #121, start hunting for it near column 7.
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 #121?
Puzzle #121 is rated hard. 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 #121 have only one solution?
Yes. Every puzzle in the 10x10 collection, including #121, 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 #121?
Start with the slate region — the most constrained color on this board, with 1 square spanning rows 5–5 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.