7x7 Queens Puzzle #140

Difficulty: medium. 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 11-step reasoning is written out below.

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Step-by-step solution for puzzle #140

What follows is a complete solve of puzzle #140, worked the way a person works it rather than the way a computer does. It runs to 11 deductions — 7 queen placements and 4 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 confined to one line, pairing argument on rows, pairing argument on columns, last square in a region.

Show the full walkthrough (5 steps) — contains the solution

  1. Step 1. Blue cannot leave row 3 — every one of its open squares (2 open squares) sits there. That row belongs to blue, so every square on it owned by another colour is dead: 9 squares in all.

  2. Step 2. All of the mint 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: 7 squares come off.

  3. Step 3. Mint, slate, red, and pale yellow are now boxed into rows 4–7 and nothing else. That is 4 colours needing 4 rows, so they fill those rows between them — which locks every remaining colour out and removes 2 squares at once.

    Puzzle #140 after step 3: 0 of 7 queens placed
    The board after step 3 — 0 of 7 queens placed.

  4. Step 4. Here is the pairing argument. The indigo and pale yellow regions have no open squares outside columns 1 and 3 — 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.

  5. Step 5. 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: blue has one square left, so its queen takes row 3, column 4; red has one square left, so its queen takes row 6, column 5; pale yellow has one square left, so its queen takes row 7, column 1; indigo has one square left, so its queen takes row 1, column 3; slate has one square left, so its queen takes row 5, column 7; amber has one square left, so its queen takes row 2, column 6; mint has one square left, so its queen takes row 4, column 2. 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 4, column 2.

    Puzzle #140 after step 5: 7 of 7 queens placed
    The board after step 5 — 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 #140

This is puzzle number 140 in the 7x7 collection, rated medium. 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 #140, 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 #140 are: blue (2 squares), mint (3 squares), red (7 squares), slate (8 squares), pale yellow (8 squares), indigo (9 squares), amber (12 squares). An even split would hand every color 7 squares, so the blue region is running 5 squares under that average while the amber region is the roomiest area on the grid with 12 squares.

Every color takes one queen and one only, whatever its size — which is why the 10 square difference between blue and amber 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 1×2 box that encloses it, while red is the most scattered at 58% of its 4×3 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 #140 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.

Blue occupies 2 squares within rows 3–3, columns 3–4. It never leaves row 3, so its queen is locked to that row.

Mint: 3 squares across rows 4–5, columns 2–3.

Red covers 7 squares, spanning 4 rows and 3 columns (rows 4–7, columns 3–5).

Slate occupies 8 squares within rows 5–7, columns 5–7.

Pale yellow: 8 squares across rows 4–7, columns 1–3.

Indigo covers 9 squares, spanning 3 rows and 4 columns (rows 1–3, columns 1–4).

Amber occupies 12 squares within rows 1–4, columns 4–7.

Where to start on this board

Open on the tightest color. Here that is blue, confined to 2 squares between rows 3 and 3 and columns 3 and 4. Its queen has nowhere else to go, which means those 1 row and 2 columns carry more constraint than anywhere else on #140.

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. 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 #140

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

Those lines matter because a row touched by few colors gives its queen away sooner. If a row meets only 2 colors, then one of those regions has to own it, and every other region is competing for the remaining 6 rows. By contrast row 5 meets 4 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.

With seven queens to seat and seven regions competing over 49 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 7x7 boards quickly and players who stall halfway.

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 #140 is rated medium

What earns a board its rating is the shape of its opening. Here that opening offers one single-line region, a 2-square smallest color, a tightest row touched by 2 colors, and a region-size spread of 10. Lopsided grids with flat regions fall quickly; balanced ones like this take more patience.

The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a medium board from the rest.

Stuck on #140 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 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 #140 is row 1.

As a sanity check rather than a spoiler: the finished solution to #140 puts 3 queens on the outer border and 4 queens in the interior, 1 of them in an actual 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

  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 #140?

Puzzle #140 is rated medium. 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 #140 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #140, 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 #140?

Start with the blue region — the most constrained color on this board, with 2 squares spanning rows 3–3 and columns 3–4. Narrowing the smallest region first removes the most squares elsewhere on the grid.