8x8 Queens Puzzle #112
Difficulty: medium. Place 8 queens on this 8x8 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 18-step reasoning is written out below.
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Step-by-step solution for puzzle #112
What follows is a complete solve of puzzle #112, worked the way a person works it rather than the way a computer does. It runs to 18 deductions — 8 queen placements and 10 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 columns.
Techniques this board needs: region shadow, region confined to one line, line confined to one region, pairing argument on columns, pairing argument on rows, last square in a region, last square in a column.
Show the full walkthrough (11 steps) — contains the solution
Step 1. 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 2. Red is down to 4 candidates in rows 6–8, columns 1 and 2, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 3. Pale yellow cannot leave column 3 — 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: 9 squares in all.
Step 4. All of the red 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: 5 squares come off.
The board after step 4 — 0 of 8 queens placed. Step 5. Blue cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to blue, so every square in it owned by another colour is dead: 4 squares in all.
Step 6. The indigo region has 4 squares left, spread across rows 1 and 2, columns 4–7. You do not need to know which of them takes the queen: every one of those options attacks the same 3 squares, so they can be crossed out now.
Step 7. Row 1 has only indigo squares left (3 open squares). The row needs a queen and only indigo can provide one, which pins that colour to this row and kills its squares elsewhere: 1 square.
The board after step 7 — 0 of 8 queens placed. Step 8. Here is the pairing argument. The indigo, mint, and slate regions have no open squares outside columns 4–6 — 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: 5 squares come off in one stroke.
Step 9. The khaki region has 3 squares left, spread across rows 7 and 8, columns 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 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: 3 squares come off in one stroke.
Step 11. 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: red has one square left, so its queen takes row 6, column 1; blue has one square left, so its queen takes row 3, column 2; mint has one square left, so its queen takes row 4, column 6; column 5 is down to one square, putting a queen on row 1, column 5; slate has one square left, so its queen takes row 5, column 4; amber has one square left, so its queen takes row 2, column 8; khaki has one square left, so its queen takes row 8, column 7; pale yellow has one square left, so its queen takes row 7, column 3. 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 3.
The board after step 11 — 8 of 8 queens placed.
That completes the board: 8 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 8x8 Queens puzzle #112
This is puzzle number 112 in the 8x8 collection, rated medium. The grid holds 64 squares split between eight colored regions, and a finished board carries eight queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
There is exactly one way to finish #112, 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 eight regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #112 are: pale yellow (2 squares), mint (3 squares), red (4 squares), blue (7 squares), khaki (9 squares), indigo (10 squares), amber (12 squares), slate (17 squares). An even split would hand every color 8 squares, so the pale yellow region is running 6 squares under that average while the slate region is the roomiest area on the grid with 17 squares.
Every color takes one queen and one only, whatever its size — which is why the 15 square difference between pale yellow and slate 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 pale yellow region is the most compact color here, filling 100% of the 2×1 box that encloses it, while khaki is the most scattered at 45% of its 4×5 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.
Pale yellow covers 2 squares, spanning 2 rows and 1 column (rows 7–8, columns 3–3). It sits wholly inside column 3, so its queen must be in that column.
Mint occupies 3 squares within rows 3–4, columns 5–6.
Red: 4 squares across rows 6–8, columns 1–2.
Blue covers 7 squares, spanning 5 rows and 2 columns (rows 1–5, columns 1–2).
Khaki occupies 9 squares within rows 5–8, columns 4–8.
Indigo: 10 squares across rows 1–3, columns 2–7.
Amber covers 12 squares, spanning 7 rows and 2 columns (rows 1–7, columns 7–8).
Slate occupies 17 squares within rows 2–7, columns 2–5.
Where to start on this board
The most constrained color is where a solve should begin, and on #112 that is pale yellow with 2 squares. It reaches from row 7 to row 8 and from column 3 to column 3 — 2 rows 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 pale yellow stays inside column 3, its queen belongs to that column and no other region may use it. Cross out the rest of column 3 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.
Work outward to the other flat regions next. Mint 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 #112
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 8 are the narrowest, touched by only 3 colors. Columns 1 and 8 are the tightest columns, with 2 colors.
The fewer colors a line touches, the sooner it surrenders its queen. A row meeting just 3 colors has to be claimed by one of them, which pushes every remaining region into the other 7 rows. Row 7, meeting 5 colors, is the opposite case — the busiest line on #112 and the last one you should expect to crack.
Working through the middle game
The middle of #112 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.
The whole middle game reduces to two sweeps. Sweep the eight colors looking for one with a single square left; sweep the eight rows and eight columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 8x8 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 #112 is rated medium
Ratings track how generous the opening is. On #112 you get one region pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 3 colors, and a 15-square gap between the largest and smallest regions. Those four numbers largely explain the medium label.
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 #112 despite the medium 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 8x8
At 8x8 the whole grid stays within a single glance, which makes systematic scanning practical. Read the rows in order and count the colors in each — a row touched by two colors is far more constrained than one touched by five, and that is usually where the next forced move hides. The board is small enough that re-scanning from the top after each placement costs almost nothing.
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 #112 is column 1.
Here is a shape check that gives nothing away: when #112 is solved, 4 queens sit on the outer edge of the grid and 4 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 8x8 Queens puzzle #112?
Puzzle #112 is rated medium. Its smallest color region holds 2 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 8x8 puzzle #112 have only one solution?
Yes. Every puzzle in the 8x8 collection, including #112, has been verified to have exactly one valid arrangement of eight queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #112?
Start with the pale yellow region — the most constrained color on this board, with 2 squares spanning rows 7–8 and columns 3–3. Narrowing the smallest region first removes the most squares elsewhere on the grid.