12x12 Queens Puzzle #16
Difficulty: easy. Place 12 queens on this 12x12 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 #16
What follows is a complete solve of puzzle #16, worked the way a person works it rather than the way a computer does. It runs to 19 deductions — 12 queen placements and 7 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.
Show the full walkthrough (16 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: indigo has one square left, so its queen takes row 1, column 5; red has one square left, so its queen takes row 6, column 1; teal has one square left, so its queen takes row 11, column 8; 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.
Step 2. Pale yellow is down to 3 candidates in rows 7 and 10, columns 10 and 11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 3. All of the rose region's remaining squares (2 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
Step 4. Sea green cannot leave column 12 — 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: 5 squares in all.
Step 5. The slate region is down to one open square, at row 5, column 10. Every colour holds exactly one queen, so that square is forced. Row 5 and column 10 are now spoken for, along with the squares touching that queen.
The board after step 5 — 5 of 12 queens placed. Step 6. Only row 12, column 6 is still open in rose, and since each colour must take a queen, that settles it. That clears row 12, column 6, the rest of the rose region and every square adjacent to it — diagonals included.
Step 7. All of the pale yellow region's remaining squares (2 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 2 squares come off.
Step 8. Blue cannot leave column 7 — every one of its open squares (3 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 2 squares in all.
Step 9. Here is the pairing argument. The khaki and sea green regions have no open squares outside rows 8 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: 4 squares come off in one stroke.
Step 10. Only row 7, column 11 is still open in pale yellow, and since each colour must take a queen, that settles it. That clears row 7, column 11, the rest of the pale yellow region and every square adjacent to it — diagonals included.
Step 11. The sea green region is down to one open square, at row 10, column 12. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 12, the remaining sea green squares, and the ring of neighbours around row 10, column 12.
The board after step 11 — 8 of 12 queens placed. Step 12. Only row 8, column 4 is still open in khaki, and since each colour must take a queen, that settles it. Everything else in row 8, in column 4, in khaki and in the ring around that square is now dead.
Step 13. 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 14. Only row 4, column 2 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 2, the rest of the mint region and every square adjacent to it — diagonals included.
Step 15. The amber region is down to one open square, at row 2, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 3, the remaining amber squares, and the ring of neighbours around row 2, column 3.
Step 16. Only row 3, column 7 is still open in blue, and since each colour must take a queen, that settles it. Everything else in row 3, in column 7, in blue and in the ring around that square is now dead.
The board after step 16 — 12 of 12 queens placed.
That completes the board: 12 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 12x12 Queens puzzle #16
This is puzzle number 16 in the 12x12 collection, rated easy. The grid holds 144 squares split between twelve colored regions, and a finished board carries twelve queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Puzzle #16 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #16 are: indigo (1 square), red (1 square), teal (1 square), orange (4 squares), slate (8 squares), sea green (10 squares), pale yellow (11 squares), rose (12 squares), amber (13 squares), khaki (22 squares), mint (27 squares), blue (34 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the blue region is the roomiest area on the grid with 34 squares.
Every color takes one queen and one only, whatever its size — which is why the 33 square difference between indigo and blue 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.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 1×1 box that encloses it, while amber is the most scattered at 52% of its 5×5 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
Region by region
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo covers 1 square, spanning 1 row and 1 column (rows 1–1, columns 5–5). It never leaves row 1, so its queen is locked to that row.
Red occupies 1 square within rows 6–6, columns 1–1. It never leaves row 6, so its queen is locked to that row.
Teal: 1 square across rows 11–11, columns 8–8. It never leaves row 11, so its queen is locked to that row.
Orange covers 4 squares, spanning 3 rows and 2 columns (rows 8–10, columns 8–9).
Slate occupies 8 squares within rows 5–7, columns 8–12.
Sea green: 10 squares across rows 8–12, columns 10–12.
Pale yellow covers 11 squares, spanning 5 rows and 4 columns (rows 6–10, columns 8–11).
Rose occupies 12 squares within rows 10–12, columns 5–10.
Amber: 13 squares across rows 1–5, columns 2–6.
Khaki covers 22 squares, spanning 7 rows and 5 columns (rows 6–12, columns 3–7).
Mint occupies 27 squares within rows 1–12, columns 1–4.
Blue: 34 squares across rows 1–6, columns 5–12.
Where to start on this board
The most constrained color is where a solve should begin, and on #16 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 5 to column 5 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. 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. Red and teal 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 #16
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 2, 3 and 4 are the narrowest, touched by only 3 colors. Columns 1 and 2 are the tightest columns, 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 11 rows. Row 10 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #16 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.
Two checks, run alternately, will carry you through the 144 squares of this grid. First: has any of the twelve regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 12x12 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 #16 is rated easy
Ratings track how generous the opening is. On #16 you get six regions pinned to a single row or column, a smallest color of 1 square, a tightest row crossed by 3 colors, and a 33-square gap between the largest and smallest regions. Those four numbers largely explain the easy label.
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.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
Techniques that work well at 12x12
At 12x12 you are managing 144 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
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 #16 is column 1.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 8 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 12x12 Queens puzzle #16?
Puzzle #16 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #16 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #16, has been verified to have exactly one valid arrangement of twelve queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #16?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.