7x7 Queens Puzzle #113

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 12-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 12 deductions — 7 queen placements and 5 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, pairing argument on columns, last square in a region.

Show the full walkthrough (6 steps) — contains the solution

  1. Step 1. Wherever amber ends up among its 6 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.

  2. Step 2. All of the blue 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.

  3. Step 3. Slate cannot leave row 5 — every one of its open squares (2 open squares) sits there. That row belongs to slate, so every square on it owned by another colour is dead: 8 squares in all.

    Puzzle #113 after step 3: 0 of 7 queens placed
    The board after step 3 — 0 of 7 queens placed.

  4. Step 4. All of the pale yellow region's remaining squares (2 open squares) lie in row 7. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 4 squares come off.

  5. Step 5. Slate and pale yellow are now boxed into columns 6 and 7 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 5 squares at once.

  6. Step 6. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: mint has one square left, so its queen takes row 4, column 5; amber has one square left, so its queen takes row 2, column 4; indigo has one square left, so its queen takes row 1, column 2; blue has one square left, so its queen takes row 3, column 1; slate has one square left, so its queen takes row 5, column 7; red has one square left, so its queen takes row 6, column 3; 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.

    Puzzle #113 after step 6: 7 of 7 queens placed
    The board after step 6 — 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 #113

Puzzle #113 is one of the hard-rated boards in the 7x7 set. Seven colors divide its 49 squares, and you win by placing seven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.

Like every board on this site, #113 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.

How the colors divide this board

From smallest to largest, the regions on puzzle #113 are: slate (2 squares), blue (3 squares), pale yellow (3 squares), amber (6 squares), mint (7 squares), indigo (12 squares), red (16 squares). An even split would hand every color 7 squares, so the slate region is running 5 squares under that average while the red region is the roomiest area on the grid with 16 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 slate, with 2 squares, is under far more pressure than red with 16. A 14-square gap like this one rewards players who start where the choices are fewest.

Shape matters as much as size. The blue region is the most compact color here, filling 100% of the 3×1 box that encloses it, while amber is the most scattered at 50% of its 3×4 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

Below is every color on #113 with its exact reach, smallest region first. It is worth a slow read: the spans reveal which rows and columns several regions are fighting over.

Slate occupies 2 squares within rows 5–5, columns 6–7. It never leaves row 5, so its queen is locked to that row.

Blue: 3 squares across rows 2–4, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.

Pale yellow covers 3 squares, spanning 2 rows and 2 columns (rows 6–7, columns 6–7).

Amber occupies 6 squares within rows 1–3, columns 4–7.

Mint: 7 squares across rows 2–4, columns 5–7.

Indigo covers 12 squares, spanning 4 rows and 4 columns (rows 1–4, columns 1–4).

Red occupies 16 squares within rows 5–7, columns 1–6.

Where to start on this board

Slate is the natural opening on #113: it is the smallest region on the grid at 2 squares, boxed into rows 5–5 and columns 6–7. Since one of those squares must hold a queen, that block of 1 row and 2 columns is the most contested part of the board.

Because slate never leaves row 5, its queen must be in that row — and that instantly settles row 5 for everyone else. Sweep along it and X out every square that is not slate. 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. Pale yellow 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 #113

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 5, 6 and 7 are the narrowest, touched by only 2 colors. Columns 2 and 3 are the tightest columns, with 2 colors.

A line with few colors is a line close to resolving. Where only 2 colors appear, one of them owns the row outright and the others must fit into the remaining 6 rows. Row 3 is the crowded extreme at 4 colors — worth revisiting later, not now.

Working through the middle game

After the first few queens the puzzle becomes bookkeeping. A placed queen wipes its row, its column and every square it touches — as many as eight neighbours — so the space collapses quickly. Record those eliminations on the board instead of carrying them in your head, and the next forced move tends to announce itself.

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.

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 hard

What earns a board its rating is the shape of its opening. Here that opening offers two single-line regions, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 14. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

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.

If #113 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.

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.

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 #113, start hunting for it near row 1.

A non-spoiling distribution check: solved, #113 carries 4 queens on the perimeter against 3 queens inside, with the corners empty. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.

How to play Queens

  1. The board is an N×N grid divided into N colored regions.
  2. Place exactly one queen (♛) in each row, each column, and each colored region.
  3. No two queens may touch each other — not horizontally, vertically, or diagonally.
  4. 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.
  5. 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 #113?

Puzzle #113 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 2 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.

Does 7x7 puzzle #113 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #113, 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 #113?

Start with the slate region — the most constrained color on this board, with 2 squares spanning rows 5–5 and columns 6–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.