12x12 Queens Puzzle #113
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 25-step reasoning is written out below.
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Step-by-step solution for puzzle #113
What follows is a complete solve of puzzle #113, worked the way a person works it rather than the way a computer does. It runs to 25 deductions — 12 queen placements and 13 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, line confined to one region, last square in a column.
Show the full walkthrough (18 steps) — contains the solution
Step 1. Only row 1, column 5 is still open in indigo, and since each colour must take a queen, that settles it. That clears row 1, column 5, the rest of the indigo region and every square adjacent to it — diagonals included.
Step 2. Sea green is down to 6 candidates in rows 10–12, columns 8–11, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about sea green itself.
Step 3. The teal region has 4 squares left, spread across rows 11 and 12, columns 10–12. 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. All of the rose region's remaining squares (3 open squares) lie in row 12. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 9 squares come off.
Step 5. Only row 11, column 12 is still open in teal, and since each colour must take a queen, that settles it. That clears row 11, column 12, the rest of the teal region and every square adjacent to it — diagonals included.
The board after step 5 — 2 of 12 queens placed. Step 6. The sea green region is down to one open square, at row 10, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 10 and column 9, the remaining sea green squares, and the ring of neighbours around row 10, column 9.
Step 7. Only row 9, column 4 is still open in orange, and since each colour must take a queen, that settles it. Everything else in row 9, in column 4, in orange and in the ring around that square is now dead.
Step 8. Pale yellow is down to 3 candidates in rows 7 and 8, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. The khaki region has 5 squares left, spread across rows 7 and 8, columns 3, 6, 7, and 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 9 — 4 of 12 queens placed. Step 10. Here is the pairing argument. The pale yellow and khaki regions have no open squares outside rows 7 and 8 — 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: 5 squares come off in one stroke.
Step 11. Red is down to 3 candidates in rows 5 and 6, columns 6 and 7, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about red itself.
Step 12. The khaki region has 4 squares left, spread across rows 7 and 8, columns 3, 6, 7, and 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.
Step 13. Pale yellow cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to pale yellow, so every square in it owned by another colour is dead: 6 squares in all.
Step 14. Slate is down to 3 candidates in rows 5 and 6, columns 2 and 3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about slate itself.
The board after step 14 — 4 of 12 queens placed. Step 15. Slate and red are now boxed into rows 5 and 6 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 6 squares at once.
Step 16. Wherever mint 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 17. Column 11 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: 5 squares.
Step 18. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: column 10 is down to one square, putting a queen on row 4, column 10; amber has one square left, so its queen takes row 2, column 11; column 8 is down to one square, putting a queen on row 8, column 8; pale yellow has one square left, so its queen takes row 7, column 1; column 7 is down to one square, putting a queen on row 6, column 7; slate has one square left, so its queen takes row 5, column 2; rose has one square left, so its queen takes row 12, column 3; blue has one square left, so its queen takes row 3, 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 3, column 6.
The board after step 18 — 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 #113
Puzzle #113 is one of the easy-rated boards in the 12x12 set. Twelve colors divide its 144 squares, and you win by placing twelve 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 #113, 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 twelve regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #113 are: indigo (1 square), rose (3 squares), teal (4 squares), pale yellow (6 squares), sea green (6 squares), slate (9 squares), red (10 squares), orange (15 squares), amber (16 squares), mint (19 squares), blue (25 squares), khaki (30 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the khaki region is the roomiest area on the grid with 30 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while khaki has 30 to spare, a gap of 29. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #113, packing 100% of its 1×1 bounding box, whereas khaki straggles across its 7×11 box at just 39%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
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.
Indigo occupies 1 square within rows 1–1, columns 5–5. It never leaves row 1, so its queen is locked to that row.
Rose: 3 squares across rows 12–12, columns 1–3. It never leaves row 12, so its queen is locked to that row.
Teal covers 4 squares, spanning 2 rows and 3 columns (rows 11–12, columns 10–12).
Pale yellow occupies 6 squares within rows 7–10, columns 1–2.
Sea green: 6 squares across rows 10–12, columns 8–11.
Slate covers 9 squares, spanning 3 rows and 5 columns (rows 4–6, columns 1–5).
Red occupies 10 squares within rows 5–8, columns 6–10.
Orange: 15 squares across rows 9–12, columns 1–6.
Amber covers 16 squares, spanning 4 rows and 6 columns (rows 1–4, columns 7–12).
Mint occupies 19 squares within rows 3–8, columns 8–12.
Blue: 25 squares across rows 1–5, columns 1–7.
Khaki covers 30 squares, spanning 7 rows and 11 columns (rows 6–12, columns 2–12).
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 1 square between rows 1 and 1 and columns 5 and 5. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #113.
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. Rose 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 #113
A quick way to find the next move is to count how many different colors touch each line. On this board, row 2 is the narrowest, touched by only 2 colors. Columns 4, 6, 7 and 12 are the tightest columns, with 4 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 2 colors has to be claimed by one of them, which pushes every remaining region into the other 11 rows. Row 12, meeting 5 colors, is the opposite case — the busiest line on #113 and the last one you should expect to crack.
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 twelve colors looking for one with a single square left; sweep the twelve rows and twelve columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 12x12 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 #113 is rated easy
The easy label comes out of four measurable things about #113: three of its regions are locked to one row or column, its smallest color spans 1 square, its least crowded row meets 2 colors, and its largest and smallest regions differ by 29. Boards that score tight on those measures give up their first queen fast.
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 easy board from the rest.
Stuck on #113 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 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.
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 2 on this board.
Here is a shape check that gives nothing away: when #113 is solved, 4 queens sit on the outer edge of the grid and 8 sit inside it, and no queen lands in a corner. A work-in-progress board that clusters its queens in one quadrant while leaving others bare is usually carrying a mistake.
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 #113?
Puzzle #113 is rated easy. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #113 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #113, 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 #113?
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.