7x7 Queens Puzzle #172

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.

Previous puzzle · All 7x7 puzzles · Next puzzle

Step-by-step solution for puzzle #172

What follows is a complete solve of puzzle #172, 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 line confined to one region.

Techniques this board needs: last square in a region, region confined to one line, region shadow, line confined to one region, last square in a column.

Show the full walkthrough (7 steps) — contains the solution

  1. Step 1. Only row 4, column 2 is still open in mint, and since each colour must take a queen, that settles it. That clears row 4, column 2, the rest of the mint region and every square adjacent to it — diagonals included.

  2. Step 2. The blue region is down to one open square, at row 3, column 7. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 3 and column 7, the remaining blue squares, and the ring of neighbours around row 3, column 7.

  3. Step 3. Amber cannot leave row 2 — every one of its open squares (2 open squares) sits there. That row belongs to amber, so every square on it owned by another colour is dead: 4 squares in all.

  4. Step 4. All of the red region's remaining squares (2 open squares) lie in column 1. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.

    Puzzle #172 after step 4: 2 of 7 queens placed
    The board after step 4 — 2 of 7 queens placed.

  5. Step 5. Pale yellow is down to 3 candidates in rows 6 and 7, columns 5 and 6, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.

  6. Step 6. Every open square left in row 5 (3 open squares) belongs to the slate region. Since that row must contain a queen, slate is the colour that supplies it — so slate squares anywhere else on the board are out, 3 squares in all.

  7. Step 7. 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: column 3 is down to one square, putting a queen on row 1, column 3; amber has one square left, so its queen takes row 2, column 5; pale yellow has one square left, so its queen takes row 7, column 6; slate has one square left, so its queen takes row 5, column 4; red has one square left, so its queen takes row 6, 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 6, column 1.

    Puzzle #172 after step 7: 7 of 7 queens placed
    The board after step 7 — 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 #172

This is puzzle number 172 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 #172, 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 #172 are: mint (1 square), blue (2 squares), pale yellow (4 squares), indigo (9 squares), amber (9 squares), red (9 squares), slate (15 squares). An even split would hand every color 7 squares, so the mint region is running 6 squares under that average while the slate region is the roomiest area on the grid with 15 squares.

Every color takes one queen and one only, whatever its size — which is why the 14 square difference between mint 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 blue is the most solid, filling 100% of its 2×1 box, and amber the most diffuse at 45% of its 4×5 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 #172, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.

Mint covers 1 square, spanning 1 row and 1 column (rows 4–4, columns 2–2). It never leaves row 4, so its queen is locked to that row.

Blue occupies 2 squares within rows 3–4, columns 7–7. It sits wholly inside column 7, so its queen must be in that column.

Pale yellow: 4 squares across rows 6–7, columns 5–7.

Indigo covers 9 squares, spanning 3 rows and 6 columns (rows 1–3, columns 1–6).

Amber occupies 9 squares within rows 1–4, columns 3–7.

Red: 9 squares across rows 2–7, columns 1–2.

Slate covers 15 squares, spanning 5 rows and 5 columns (rows 3–7, columns 3–7).

Where to start on this board

Open on the tightest color. Here that is mint, confined to 1 square between rows 4 and 4 and columns 2 and 2. Its queen has nowhere else to go, which means those 1 row and 1 column carry more constraint than anywhere else on #172.

Because mint never leaves row 4, its queen must be in that row — and that instantly settles row 4 for everyone else. Sweep along it and X out every square that is not mint. 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. Blue 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 #172

A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1 and 5 are the narrowest, touched by only 2 colors. Column 1 is the tightest column, 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 4 is the crowded extreme at 5 colors — worth revisiting later, not now.

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 49 squares of this grid. First: has any of the seven regions been reduced to a single open square? Second: has any row or column? Either answer forces a queen. Players who stall on 7x7 boards almost always run one of those checks and forget the other.

Resist the plausible-looking queen. On #172 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 #172 is rated medium

The rating reflects how much the board concedes up front. Puzzle #172 has three regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 2 colors, and the gap between its largest and smallest regions is 14 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.

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 #172 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 #172 is row 1.

A non-spoiling distribution check: solved, #172 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.

Full rules · Strategy library · Puzzle statistics

Questions about this puzzle

How hard is 7x7 Queens puzzle #172?

Puzzle #172 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 2 colors, and three of its regions are confined to a single row or column, which gives you a forced move early.

Does 7x7 puzzle #172 have only one solution?

Yes. Every puzzle in the 7x7 collection, including #172, 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 #172?

Start with the mint region — the most constrained color on this board, with 1 square spanning rows 4–4 and columns 2–2. Narrowing the smallest region first removes the most squares elsewhere on the grid.