12x12 Queens Puzzle #106

Difficulty: medium. 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 42-step reasoning is written out below.

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Step-by-step solution for puzzle #106

What follows is a complete solve of puzzle #106, worked the way a person works it rather than the way a computer does. It runs to 42 deductions — 12 queen placements and 30 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: region shadow, region confined to one line, pairing argument on columns, proof by contradiction, last square in a row, line confined to one region, last square in a region, pairing argument on rows, last square in a column.

Show the full walkthrough (35 steps) — contains the solution

Step 1. Wherever orange ends up among its 7 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 2. All of the sea green region's remaining squares (3 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 10 squares come off.
Step 3. The khaki region has 5 squares left, spread across rows 7–10, columns 2 and 3. You do not need to know which of them takes the queen: every one of those options attacks the same 2 squares, so they can be crossed out now.
Step 4. Here is the pairing argument. The blue and orange regions have no open squares outside columns 11 and 12 — 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: 9 squares come off in one stroke.
Step 5. Amber is down to 3 candidates in rows 1 and 2, columns 9 and 10, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
Step 6. Here is the pairing argument. The amber, slate, and rose regions have no open squares outside columns 8–10 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 15 squares come off in one stroke.
Step 7. Wherever mint ends up among its 3 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 8. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 2. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 9. Test row 1, column 3 for indigo: place a queen there, run the forced consequences through, and the teal region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
The board after step 9 — 0 of 12 queens placed.
Step 10. Now a short proof by contradiction. Suppose the indigo queen went to row 1, column 6. Follow the forced moves that would create and column 8 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 11. Test row 2, column 2 for indigo: place a queen there, run the forced consequences through, and row 5 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 12. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 3. Follow the forced moves that would create and the khaki region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 13. Test row 2, column 4 for indigo: place a queen there, run the forced consequences through, and row 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 14. Now a short proof by contradiction. Suppose the indigo queen went to row 2, column 5. Follow the forced moves that would create and row 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 15. Test row 3, column 2 for indigo: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 16. Now a short proof by contradiction. Suppose the indigo queen went to row 3, column 3. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 17. Test row 3, column 4 for indigo: place a queen there, run the forced consequences through, and the orange region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 18. Now a short proof by contradiction. Suppose the indigo queen went to row 3, column 5. Follow the forced moves that would create and the orange region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
The board after step 18 — 0 of 12 queens placed.
Step 19. Look along row 3: one square survives, row 3, column 12, which belongs to blue. A row cannot be left empty, so the queen goes there. That clears row 3, column 12, the rest of the blue region and every square adjacent to it — diagonals included.
Step 20. All of the orange region's remaining squares (3 open squares) lie in column 11. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 21. Test row 4, column 2 for indigo: place a queen there, run the forced consequences through, and row 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 22. Now a short proof by contradiction. Suppose the indigo queen went to row 4, column 3. Follow the forced moves that would create and row 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 23. Test row 4, column 4 for indigo: place a queen there, run the forced consequences through, and row 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 24. Now a short proof by contradiction. Suppose the indigo queen went to row 4, column 5. Follow the forced moves that would create and row 6 runs out of open squares entirely — impossible. So that square is ruled out for good.
Step 25. Test row 4, column 6 for indigo: place a queen there, run the forced consequences through, and row 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 26. Row 4 has a single open square left, at column 7 in the mint region. Every row needs a queen, so this one is forced. Row 4 and column 7 are now spoken for, along with the squares touching that queen.
The board after step 26 — 2 of 12 queens placed.
Step 27. Row 2 has only amber squares left (2 open squares). The row needs a queen and only amber can provide one, which pins that colour to this row and kills its squares elsewhere: 1 square.
Step 28. Every open square left in row 1 (2 open squares) belongs to the indigo region. Since that row must contain a queen, indigo is the colour that supplies it — so indigo squares anywhere else on the board are out, 3 squares in all.
Step 29. Test row 2, column 10 for amber: place a queen there, run the forced consequences through, and row 6 runs out of open squares entirely. The assumption breaks, so the square is eliminated.
Step 30. The amber region is down to one open square, at row 2, column 9. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 2 and column 9, the remaining amber squares, and the ring of neighbours around row 2, column 9.
Step 31. Sea green, teal, and rose are now boxed into rows 10–12 and nothing else. That is 3 colours needing 3 rows, so they fill those rows between them — which locks every remaining colour out and removes 7 squares at once.
Step 32. The pale yellow region is down to one open square, at row 7, column 6. Every colour holds exactly one queen, so that square is forced. Row 7 and column 6 are now spoken for, along with the squares touching that queen.
Step 33. Orange cannot leave column 11 — every one of its open squares (2 open squares) sits there. That column belongs to orange, so every square in it owned by another colour is dead: 1 square in all.
Step 34. Here is the pairing argument. The khaki and orange regions have no open squares outside rows 8 and 9 — 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 35. 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 8 is down to one square, putting a queen on row 12, column 8; teal has one square left, so its queen takes row 11, column 3; sea green has one square left, so its queen takes row 10, column 1; khaki has one square left, so its queen takes row 8, column 2; red has one square left, so its queen takes row 6, column 4; indigo has one square left, so its queen takes row 1, column 5; slate has one square left, so its queen takes row 5, column 10; orange has one square left, so its queen takes row 9, column 11. 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 11.
The board after step 35 — 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 #106

Board #106 of the 12x12 bank carries a medium rating. Its 144 squares are shared between twelve colored regions, and the solution places twelve queens: one per row, one per column, one per region, none of them side by side or corner to corner.

Puzzle #106 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 #106 are: sea green (3 squares), amber (7 squares), orange (7 squares), blue (8 squares), teal (9 squares), slate (10 squares), rose (11 squares), khaki (12 squares), mint (16 squares), pale yellow (16 squares), red (19 squares), indigo (26 squares). An even split would hand every color 12 squares, so the sea green region is running 9 squares under that average while the indigo region is the roomiest area on the grid with 26 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 sea green, with 3 squares, is under far more pressure than indigo with 26. A 23-square gap like this one rewards players who start where the choices are fewest.

Look at arrangement as well as area. Sea green is the densest region on this grid at 100% of a 3×1 frame; teal is the loosest, occupying only 50% of its 3×6 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.

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 medium board.

Sea green covers 3 squares, spanning 3 rows and 1 column (rows 10–12, columns 1–1). It sits wholly inside column 1, so its queen must be in that column.

Amber occupies 7 squares within rows 1–3, columns 9–12.

Orange: 7 squares across rows 7–12, columns 11–12.

Blue covers 8 squares, spanning 7 rows and 2 columns (rows 2–8, columns 11–12).

Teal occupies 9 squares within rows 10–12, columns 2–7.

Slate: 10 squares across rows 5–9, columns 8–11.

Rose covers 11 squares, spanning 4 rows and 4 columns (rows 9–12, columns 8–11).

Khaki occupies 12 squares within rows 3–10, columns 1–3.

Mint: 16 squares across rows 1–6, columns 6–10.

Pale yellow covers 16 squares, spanning 5 rows and 5 columns (rows 7–11, columns 4–8).

Red occupies 19 squares within rows 5–9, columns 2–7.

Indigo: 26 squares across rows 1–5, columns 1–7.

Where to start on this board

Sea green is the natural opening on #106: it is the smallest region on the grid at 3 squares, boxed into rows 10–12 and columns 1–1. Since one of those squares must hold a queen, that block of 3 rows and 1 column is the most contested part of the board.

Because sea green stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 straight away. Any color that was relying on those squares now has a visibly smaller footprint, and that shrinkage is where your second deduction comes from.

No region here is pinned to a shallow band of rows, so turn the same idea sideways: blue and orange 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 #106

A quick way to find the next move is to count how many different colors touch each line. On this board, row 1 is the narrowest, touched by only 3 colors. Columns 1 and 12 are the tightest columns, with 3 colors.

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 3 colors, then one of those regions has to own it, and every other region is competing for the remaining 11 rows. By contrast row 10 meets 6 colors — the most crowded line here, and the one least likely to resolve early. Spend your attention accordingly.

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.

Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.

Why puzzle #106 is rated medium

The medium label comes out of four measurable things about #106: one of its regions is locked to one row or column, its smallest color spans 3 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 23. Boards that score tight on those measures give up their first queen fast.

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.

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 #106, start hunting for it near row 1.

As a sanity check rather than a spoiler: the finished solution to #106 puts 4 queens on the outer border and 8 queens in the interior, with none in a corner. If your part-finished grid has queens bunched into one area while whole stretches sit empty, something probably went wrong a few moves back.

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 #106?

Puzzle #106 is rated medium. Its smallest color region holds 3 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.

Does 12x12 puzzle #106 have only one solution?

Yes. Every puzzle in the 12x12 collection, including #106, 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 #106?

Start with the sea green region — the most constrained color on this board, with 3 squares spanning rows 10–12 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.