7x7 Queens Puzzle #87
Difficulty: easy. 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 #87
What follows is a complete solve of puzzle #87, 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 pairing argument on rows.
Techniques this board needs: region confined to one line, region shadow, line confined to one region, pairing argument on rows, last square in a region.
Show the full walkthrough (13 steps) — contains the solution
Step 1. Amber cannot leave column 1 — every one of its open squares (2 open squares) sits there. That column belongs to amber, so every square in it owned by another colour is dead: 7 squares in all.
Step 2. Blue is down to 4 candidates in rows 3 and 4, columns 4–6, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about blue itself.
Step 3. The red region has 4 squares left, spread across rows 5–7, columns 2 and 3. 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 4. All of the pale yellow region's remaining squares (3 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.
The board after step 4 — 0 of 7 queens placed. Step 5. Red cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to red, so every square in it owned by another colour is dead: 3 squares in all.
Step 6. Every open square left in column 3 (3 open squares) belongs to the indigo region. Since that column must contain a queen, indigo is the colour that supplies it — so indigo squares anywhere else on the board are out, 10 squares in all.
Step 7. Column 7 has only mint squares left (3 open squares). The column needs a queen and only mint can provide one, which pins that colour to this column and kills its squares elsewhere: 1 square.
Step 8. Here is the pairing argument. The slate and red regions have no open squares outside rows 5 and 6 — 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: 2 squares come off in one stroke.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 4 queens fall out in order without any new idea being needed: mint has one square left, so its queen takes row 4, column 7; blue has one square left, so its queen takes row 3, column 4; indigo has one square left, so its queen takes row 1, column 3; 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 9 — 4 of 7 queens placed. Step 10. Wherever slate ends up among its 2 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 11. The pale yellow region is down to one open square, at row 7, column 6. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 6, the remaining pale yellow squares, and the ring of neighbours around row 7, column 6.
Step 12. Only row 5, column 5 is still open in slate, and since each colour must take a queen, that settles it. Everything else in row 5, in column 5, in slate and in the ring around that square is now dead.
Step 13. The red region is down to one open square, at row 6, column 2. Every colour holds exactly one queen, so that square is forced. Row 6 and column 2 are now spoken for, along with the squares touching that queen.
The board after step 13 — 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 #87
7x7 puzzle #87 is rated easy. Seven colored regions cover its 49 squares, and your job is to seat seven queens so that every row, column and color contains exactly one — with no queen touching another, including diagonally.
Puzzle #87 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #87 are: amber (2 squares), pale yellow (3 squares), blue (4 squares), mint (5 squares), red (7 squares), slate (8 squares), indigo (20 squares). An even split would hand every color 7 squares, so the amber region is running 5 squares under that average while the indigo region is the roomiest area on the grid with 20 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Amber is working with 2 squares while indigo has 20 to spare, a gap of 18. Attack the cramped end and the roomy regions tend to resolve themselves.
Look at arrangement as well as area. Amber is the densest region on this grid at 100% of a 2×1 frame; mint is the loosest, occupying only 63% of its 4×2 frame. The loose one reaches into more rows and columns, which is exactly why it usually settles last.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a easy board.
Amber: 2 squares across rows 1–2, columns 1–1. It sits wholly inside column 1, so its queen must be in that column.
Pale yellow covers 3 squares, spanning 1 row and 3 columns (rows 7–7, columns 4–6). It never leaves row 7, so its queen is locked to that row.
Blue occupies 4 squares within rows 3–4, columns 4–6.
Mint: 5 squares across rows 4–7, columns 6–7.
Red covers 7 squares, spanning 3 rows and 3 columns (rows 5–7, columns 1–3).
Slate occupies 8 squares within rows 4–6, columns 3–6.
Indigo: 20 squares across rows 1–4, columns 1–7.
Where to start on this board
The most constrained color is where a solve should begin, and on #87 that is amber with 2 squares. It reaches from row 1 to row 2 and from column 1 to column 1 — 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 amber stays inside column 1, its queen belongs to that column and no other region may use it. Cross out the rest of column 1 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. Pale yellow and blue 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 #87
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2 and 3 are the narrowest, touched by only 2 colors. Columns 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 4, meeting 4 colors, is the opposite case — the busiest line on #87 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.
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.
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 #87 is rated easy
The easy label comes out of four measurable things about #87: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 2 colors, and its largest and smallest regions differ by 18. Boards that score tight on those measures give up their first queen fast.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
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.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.
Here is a shape check that gives nothing away: when #87 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 #87?
Puzzle #87 is rated easy. 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 #87 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #87, 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 #87?
Start with the amber region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.