7x7 Queens Puzzle #56
Difficulty: hard. Place 7 queens on this 7x7 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 16-step reasoning is written out below.
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Step-by-step solution for puzzle #56
What follows is a complete solve of puzzle #56, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 7 queen placements and 9 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 confined to one line, region shadow, pairing argument on rows, pairing argument on columns, last square in a region, proof by contradiction, last square in a column.
Show the full walkthrough (12 steps) — contains the solution
Step 1. Indigo cannot leave column 5 — every one of its open squares (2 open squares) sits there. That column belongs to indigo, so every square in it owned by another colour is dead: 9 squares in all.
Step 2. Amber is down to 6 candidates in rows 1–3, columns 1–3, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about amber itself.
Step 3. The blue region has 3 squares left, spread across rows 1–3, columns 6 and 7. 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. 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.
The board after step 4 — 0 of 7 queens placed. Step 5. Red is down to 4 candidates in rows 5–7, columns 6 and 7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 6. Here is the pairing argument. The indigo, amber, and blue regions have no open squares outside rows 1–3 — 3 colours confined to 3 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 7. Mint cannot leave row 4 — every one of its open squares (2 open squares) sits there. That row belongs to mint, so every square on it owned by another colour is dead: 2 squares in all.
Step 8. Here is the pairing argument. The blue and red regions have no open squares outside columns 6 and 7 — 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: 2 squares come off in one stroke.
The board after step 8 — 0 of 7 queens placed. Step 9. Only row 4, column 4 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 4, the rest of the mint region and every square adjacent to it — diagonals included.
Step 10. The pale yellow region is down to one open square, at row 7, column 3. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 3, the remaining pale yellow squares, and the ring of neighbours around row 7, column 3.
Step 11. Test row 2, column 5 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 12. From here the board cascades. Keep alternating the region sweep and the line sweep and 5 queens fall out in order without any new idea being needed: indigo has one square left, so its queen takes row 1, column 5; column 2 is down to one square, putting a queen on row 5, column 2; red has one square left, so its queen takes row 6, column 7; blue has one square left, so its queen takes row 3, column 6; amber has one square left, so its queen takes row 2, column 1. 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 2, column 1.
The board after step 12 — 7 of 7 queens placed.
That completes the board: 7 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 7x7 Queens puzzle #56
This is puzzle number 56 in the 7x7 collection, rated hard. The grid holds 49 squares split between seven colored regions, and a finished board carries seven 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 #56, 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 seven regions is what sets the depth.
How the colors divide this board
From smallest to largest, the regions on puzzle #56 are: indigo (2 squares), blue (5 squares), mint (5 squares), red (6 squares), pale yellow (6 squares), amber (8 squares), slate (17 squares). An even split would hand every color 7 squares, so the indigo region is running 5 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 indigo 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.
A region's outline can constrain it more than its size does. Here indigo is the most solid, filling 100% of its 2×1 box, and slate the most diffuse at 57% of its 6×5 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
Region by region
Here is the full footprint of every color on puzzle #56, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo occupies 2 squares within rows 1–2, columns 5–5. It sits wholly inside column 5, so its queen must be in that column.
Blue: 5 squares across rows 1–3, columns 6–7.
Mint covers 5 squares, spanning 2 rows and 3 columns (rows 3–4, columns 4–6).
Red occupies 6 squares within rows 3–7, columns 6–7.
Pale yellow: 6 squares across rows 6–7, columns 3–6.
Amber covers 8 squares, spanning 3 rows and 4 columns (rows 1–3, columns 1–4).
Slate occupies 17 squares within rows 2–7, columns 1–5.
Where to start on this board
Open on the tightest color. Here that is indigo, confined to 2 squares between rows 1 and 2 and columns 5 and 5. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #56.
Because indigo stays inside column 5, its queen belongs to that column and no other region may use it. Cross out the rest of column 5 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 and pale yellow 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 #56
A quick way to find the next move is to count how many different colors touch each line. On this board, row 5 is the narrowest, touched by only 2 colors. Columns 1, 2 and 7 are the tightest columns, with 2 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 6 rows. Row 3, meeting 5 colors, is the opposite case — the busiest line on #56 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 seven colors looking for one with a single square left; sweep the seven rows and seven columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 7x7 solve from a stalled one.
Resist the plausible-looking queen. On #56 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #56 is rated hard
The rating reflects how much the board concedes up front. Puzzle #56 has one region confined to a single row or column, its smallest color holds 2 squares, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 15 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
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 #56 despite the hard 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 7x7
At 7x7 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 #56 is row 5.
Here is a shape check that gives nothing away: when #56 is solved, 4 queens sit on the outer edge of the grid and 3 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 7x7 Queens puzzle #56?
Puzzle #56 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 7x7 puzzle #56 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #56, has been verified to have exactly one valid arrangement of seven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #56?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 5–5. Narrowing the smallest region first removes the most squares elsewhere on the grid.