12x12 Queens Puzzle #68
Difficulty: hard. 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 24-step reasoning is written out below.
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Step-by-step solution for puzzle #68
What follows is a complete solve of puzzle #68, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 12 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 confined to one line, region shadow, proof by contradiction.
Show the full walkthrough (17 steps) — contains the solution
Step 1. Only row 3, column 7 is still open in blue, and since each colour must take a queen, that settles it. That clears row 3, column 7, the rest of the blue region and every square adjacent to it — diagonals included.
Step 2. The orange region is down to one open square, at row 9, column 11. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 9 and column 11, the remaining orange squares, and the ring of neighbours around row 9, column 11.
Step 3. Indigo cannot leave row 1 — every one of its open squares (2 open squares) sits there. That row belongs to indigo, so every square on it owned by another colour is dead: 9 squares in all.
Step 4. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 5 squares come off.
The board after step 4 — 2 of 12 queens placed. Step 5. Teal is down to 3 candidates in rows 11 and 12, columns 2 and 3, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about teal itself.
Step 6. Now a short proof by contradiction. Suppose the mint queen went to row 4, column 2. Follow the forced moves that would create and the rose region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 7. Test row 4, column 3 for mint: place a queen there, run the forced consequences through, and the pale yellow region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 8. Now a short proof by contradiction. Suppose the mint queen went to row 5, column 2. Follow the forced moves that would create and column 5 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 9. Test row 5, column 3 for mint: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 9 — 2 of 12 queens placed. Step 10. Wherever mint 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 11. Test row 6, column 2 for mint: place a queen there, run the forced consequences through, and column 9 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. The mint region has 2 squares left, spread across rows 4 and 7, 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 13. Test row 1, column 6 for indigo: place a queen there, run the forced consequences through, and column 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
The board after step 13 — 2 of 12 queens placed. Step 14. The indigo region is down to one open square, at row 1, column 5. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 1 and column 5, the remaining indigo squares, and the ring of neighbours around row 1, column 5.
Step 15. Only row 2, column 3 is still open in amber, and since each colour must take a queen, that settles it. Everything else in row 2, in column 3, in amber and in the ring around that square is now dead.
Step 16. All of the teal region's remaining squares (2 open squares) lie in column 2. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 4 squares come off.
Step 17. 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: mint has one square left, so its queen takes row 4, column 1; slate has one square left, so its queen takes row 5, column 12; red has one square left, so its queen takes row 6, column 10; sea green has one square left, so its queen takes row 10, column 9; rose has one square left, so its queen takes row 12, column 8; teal has one square left, so its queen takes row 11, column 2; khaki has one square left, so its queen takes row 8, column 4; pale yellow has one square left, so its queen takes row 7, 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 7, column 6.
The board after step 17 — 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 #68
This is puzzle number 68 in the 12x12 collection, rated hard. 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 #68 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 #68 are: blue (1 square), orange (1 square), teal (3 squares), amber (6 squares), indigo (7 squares), red (12 squares), rose (12 squares), mint (13 squares), sea green (17 squares), slate (21 squares), pale yellow (21 squares), khaki (30 squares). An even split would hand every color 12 squares, so the blue region is running 11 squares under that average while the khaki region is the roomiest area on the grid with 30 squares.
Every color takes one queen and one only, whatever its size — which is why the 29 square difference between blue and khaki 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 blue region is the most compact color here, filling 100% of the 1×1 box that encloses it, while red is the most scattered at 48% 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
Here is the full footprint of every color on puzzle #68, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Blue occupies 1 square within rows 3–3, columns 7–7. It never leaves row 3, so its queen is locked to that row.
Orange: 1 square across rows 9–9, columns 11–11. It never leaves row 9, so its queen is locked to that row.
Teal covers 3 squares, spanning 2 rows and 2 columns (rows 11–12, columns 2–3).
Amber occupies 6 squares within rows 1–3, columns 3–5.
Indigo: 7 squares across rows 1–3, columns 4–7.
Red covers 12 squares, spanning 5 rows and 5 columns (rows 4–8, columns 8–12).
Rose occupies 12 squares within rows 9–12, columns 6–10.
Mint: 13 squares across rows 1–7, columns 1–3.
Sea green covers 17 squares, spanning 7 rows and 5 columns (rows 6–12, columns 8–12).
Slate occupies 21 squares within rows 1–5, columns 7–12.
Pale yellow: 21 squares across rows 4–8, columns 3–9.
Khaki covers 30 squares, spanning 8 rows and 6 columns (rows 5–12, columns 1–6).
Where to start on this board
The most constrained color is where a solve should begin, and on #68 that is blue with 1 square. It reaches from row 3 to row 3 and from column 7 to column 7 — 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 blue never leaves row 3, its queen must be in that row — and that instantly settles row 3 for everyone else. Sweep along it and X out every square that is not blue. 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. Orange 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 #68
A quick way to find the next move is to count how many different colors touch each line. On this board, row 10 is the narrowest, touched by only 3 colors. Column 1 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 11 rows. Row 7 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
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.
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 #68 is rated hard
The rating reflects how much the board concedes up front. Puzzle #68 has four regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 29 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.
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 #68 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 #68?
Puzzle #68 is rated hard. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and four of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #68 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #68, 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 #68?
Start with the blue region — the most constrained color on this board, with 1 square spanning rows 3–3 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.