7x7 Queens Puzzle #36
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 13-step reasoning is written out below.
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Step-by-step solution for puzzle #36
What follows is a complete solve of puzzle #36, worked the way a person works it rather than the way a computer does. It runs to 13 deductions — 7 queen placements and 6 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, last square in a row.
Show the full walkthrough (7 steps) — contains the solution
Step 1. Indigo cannot leave column 7 — 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: 7 squares in all.
Step 2. Amber is down to 3 candidates in rows 1 and 2, columns 4 and 5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
Step 3. The slate region has 4 squares left, spread across rows 4–6, columns 1 and 2. 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. All of the red region's remaining squares (2 open squares) lie in column 6. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 5 squares come off.
The board after step 4 — 0 of 7 queens placed. Step 5. Column 3 has only pale yellow squares left (7 open squares). The column needs a queen and only pale yellow can provide one, which pins that colour to this column and kills its squares elsewhere: 5 squares.
Step 6. Here is the pairing argument. The indigo and amber regions have no open squares outside rows 1 and 2 — 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: 6 squares come off in one stroke.
Step 7. 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 2; row 4 is down to one square, putting a queen on 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 7; red has one square left, so its queen takes row 6, column 6; slate has one square left, so its queen takes row 5, column 1; pale yellow has one square left, so its queen takes row 7, column 3. 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 3.
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 #36
This is puzzle number 36 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 #36, 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 #36 are: indigo (2 squares), slate (4 squares), amber (5 squares), red (5 squares), blue (6 squares), mint (13 squares), pale yellow (14 squares). An even split would hand every color 7 squares, so the indigo region is running 5 squares under that average while the pale yellow region is the roomiest area on the grid with 14 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 indigo, with 2 squares, is under far more pressure than pale yellow with 14. A 12-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The indigo region is the most compact color here, filling 100% of the 2×1 box that encloses it, while pale yellow is the most scattered at 50% of its 7×4 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
The complete breakdown of each region on this board follows, ordered from tightest to roomiest. Skimming the spans first tells you where the board is crowded and where it has slack.
Indigo: 2 squares across rows 1–2, columns 7–7. It sits wholly inside column 7, so its queen must be in that column.
Slate covers 4 squares, spanning 3 rows and 2 columns (rows 4–6, columns 1–2).
Amber occupies 5 squares within rows 1–2, columns 4–6.
Red: 5 squares across rows 5–7, columns 6–7.
Blue covers 6 squares, spanning 3 rows and 2 columns (rows 1–3, columns 1–2).
Mint occupies 13 squares within rows 3–7, columns 4–7.
Pale yellow: 14 squares across rows 1–7, columns 1–4.
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 7 and 7. Its queen has nowhere else to go, which means those 2 rows and 1 column carry more constraint than anywhere else on #36.
Because indigo stays inside column 7, its queen belongs to that column and no other region may use it. Cross out the rest of column 7 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. Amber 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 #36
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 3, 4 and 7 are the narrowest, touched by only 3 colors. Column 3 is the tightest column, with 1 color.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 3 colors, so one of that handful must take the row and the rest are squeezed into the other 6 rows. Row 6 sits at the far end with 4 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
The middle of #36 is where careless play costs time. Every queen you place kills a row, a column and the eight surrounding squares; if you do not mark all of that immediately, you will later "discover" a move that was available several turns ago. Mark first, then look.
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 #36 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 #36 is rated hard
Ratings track how generous the opening is. On #36 you get one region pinned to a single row or column, a smallest color of 2 squares, a tightest row crossed by 3 colors, and a 12-square gap between the largest and smallest regions. Those four numbers largely explain the hard label.
Difficulty in Queens is never arithmetic and never luck. Every board has one solution and none require guessing. What varies is the length of the chain you must follow before the next queen becomes certain — one or two steps on an easy board, five or six on a hard one, sometimes with several partial eliminations held in play at once.
Stuck on #36 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.
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 #36, start hunting for it near column 3.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 4 queens in the middle, including 1 in a corner. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
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 #36?
Puzzle #36 is rated hard. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and one of its regions is confined to a single row or column, which gives you a forced move early.
Does 7x7 puzzle #36 have only one solution?
Yes. Every puzzle in the 7x7 collection, including #36, 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 #36?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 7–7. Narrowing the smallest region first removes the most squares elsewhere on the grid.