9x9 Queens Puzzle #137
Difficulty: easy. Place 9 queens on this 9x9 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 #137
What follows is a complete solve of puzzle #137, worked the way a person works it rather than the way a computer does. It runs to 16 deductions — 9 queen placements and 7 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, last square in a region, region shadow, pairing argument on rows, pairing argument on columns.
Show the full walkthrough (9 steps) — contains the solution
Step 1. Indigo cannot leave column 9 — 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: 9 squares in all.
Step 2. All of the amber region's remaining squares (2 open squares) lie in row 2. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
Step 3. Only row 1, column 9 is still open in indigo, and since each colour must take a queen, that settles it. Everything else in row 1, in column 9, in indigo and in the ring around that square is now dead.
The board after step 3 — 1 of 9 queens placed. Step 4. Wherever blue ends up among its 4 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 5. Slate is down to 4 candidates in rows 5 and 6, columns 1–3, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about slate itself.
Step 6. All of the khaki region's remaining squares (2 open squares) lie in row 8. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 10 squares come off.
The board after step 6 — 1 of 9 queens placed. Step 7. Pale yellow and orange are now boxed into rows 7 and 9 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 1 square at once.
Step 8. Here is the pairing argument. The blue, slate, and pale yellow regions have no open squares outside columns 1–3 — 3 colours confined to 3 columns. Between them they must take one column each, in some order, so every other colour is shut out of all of them: 3 squares come off in one stroke.
Step 9. From here the board cascades. Keep alternating the region sweep and the line sweep and 8 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 4; mint has one square left, so its queen takes row 4, column 5; blue has one square left, so its queen takes row 3, column 1; orange has one square left, so its queen takes row 9, column 6; pale yellow has one square left, so its queen takes row 7, column 2; slate has one square left, so its queen takes row 5, column 3; khaki has one square left, so its queen takes row 8, column 8; red has one square left, so its queen takes row 6, column 7. 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 7.
The board after step 9 — 9 of 9 queens placed.
That completes the board: 9 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 9x9 Queens puzzle #137
Puzzle #137 is one of the easy-rated boards in the 9x9 set. Nine colors divide its 81 squares, and you win by placing nine queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
As with every puzzle in this collection, #137 has a single verified solution and needs no guesswork. Each queen follows from something already visible. What the board controls is tempo — how soon the first forced move shows up and how far apart the rest are spaced — and that is a direct consequence of how these colors are cut.
How the colors divide this board
From smallest to largest, the regions on puzzle #137 are: indigo (2 squares), amber (2 squares), khaki (4 squares), slate (5 squares), pale yellow (8 squares), blue (10 squares), orange (10 squares), mint (17 squares), red (23 squares). An even split would hand every color 9 squares, so the indigo region is running 7 squares under that average while the red region is the roomiest area on the grid with 23 squares.
Every color takes one queen and one only, whatever its size — which is why the 21 square difference between indigo and red 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 indigo is the most solid, filling 100% of its 2×1 box, and mint the most diffuse at 40% of its 6×7 box. Solid blocks give clean eliminations; diffuse ones spread across the board and stay open longer.
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.
Indigo occupies 2 squares within rows 1–2, columns 9–9. It sits wholly inside column 9, so its queen must be in that column.
Amber: 2 squares across rows 2–2, columns 3–4. It never leaves row 2, so its queen is locked to that row.
Khaki covers 4 squares, spanning 2 rows and 3 columns (rows 8–9, columns 7–9).
Slate occupies 5 squares within rows 5–6, columns 1–3.
Pale yellow: 8 squares across rows 7–9, columns 1–3.
Blue covers 10 squares, spanning 5 rows and 4 columns (rows 1–5, columns 1–4).
Orange occupies 10 squares within rows 7–9, columns 3–8.
Mint: 17 squares across rows 1–6, columns 2–8.
Red covers 23 squares, spanning 6 rows and 5 columns (rows 3–8, columns 5–9).
Where to start on this board
The most constrained color is where a solve should begin, and on #137 that is indigo with 2 squares. It reaches from row 1 to row 2 and from column 9 to column 9 — 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 indigo stays inside column 9, its queen belongs to that column and no other region may use it. Cross out the rest of column 9 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, slate and khaki 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 #137
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 3, 4, 6, 7 and 9 are the narrowest, touched by only 3 colors. Columns 1, 5, 6 and 9 are the tightest columns, with 3 colors.
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 8 rows. Row 8 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
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.
The whole middle game reduces to two sweeps. Sweep the nine colors looking for one with a single square left; sweep the nine rows and nine columns looking for the same. Each sweep feeds the next, and switching between them deliberately — rather than drifting — is what separates a fast 9x9 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 #137 is rated easy
The easy label comes out of four measurable things about #137: two of its regions are locked to one row or column, its smallest color spans 2 squares, its least crowded row meets 3 colors, and its largest and smallest regions differ by 21. Boards that score tight on those measures give up their first queen fast.
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 easy board from the rest.
A board that feels above its rating is nearly always a board with a missing X. Audit every queen on the grid: full row cleared, full column cleared, all eight touching squares cleared. Diagonals are the common omission, and missing one is far more often the cause of a stall than any genuine difficulty in the puzzle.
Techniques that work well at 9x9
At 9x9 a full re-scan after every move starts to get slow, so work locally instead. Once a queen goes down, check only the regions, rows and columns that placement actually touched — that is where new information appeared. Fall back to a complete sweep only when the local pass turns up nothing, and your solve times will drop noticeably.
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 #137 is row 1.
Use this as a rough check on your progress: the completed board has 3 queens around the border and 6 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 9x9 Queens puzzle #137?
Puzzle #137 is rated easy. Its smallest color region holds 2 squares, its tightest row is touched by 3 colors, and two of its regions are confined to a single row or column, which gives you a forced move early.
Does 9x9 puzzle #137 have only one solution?
Yes. Every puzzle in the 9x9 collection, including #137, has been verified to have exactly one valid arrangement of nine queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #137?
Start with the indigo region — the most constrained color on this board, with 2 squares spanning rows 1–2 and columns 9–9. Narrowing the smallest region first removes the most squares elsewhere on the grid.